Optimize Designs for Collision Safety - Physics

A 70 kg passenger in a car experiences a frontal collision and is brought from $v=20\ \text{m/s}$ to rest by an airbag. To minimize injury, the average force on the passenger must be kept below $F_{\max}=3500\ \text{N}$ . The airbag system can be tuned by changing the effective collision time $\Delta t$ , but packaging limits require $\Delta t \le 0.60\ \text{s}$ . What is the minimum collision time $\Delta t_{\min}$ required to keep the average force below the limit, and is it feasible within the packaging constraint? (Use $F_{\text{avg}}=\Delta p/\Delta t$ .)
Answer: $\Delta t_{\min}=0.40\ \text{s}$ ; feasible because $0.40\le 0.60$ This question tests understanding of optimizing collision safety designs using impulse-momentum or work-energy to minimize forces while meeting constraints. To minimize force for a given momentum change Δp = mv, solve F_max = Δp/Δt_min for minimum time: Δt_min = mv/F_max, or for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 70 kg impacting at v = 20 m/s, the momentum change is Δp = mv = 70*20 = 1400 kg⋅m/s. To keep force below F_max = 3500 N, the minimum collision time is Δt_min = Δp/F_max = 1400/3500 = 0.4 s. The packaging limit allows up to Δt = 0.60 s which is greater than this minimum, so F_avg can be kept < threshold. Choice B is correct because it correctly calculates Δt_min using the appropriate formula and properly evaluates against both safety goal and constraints. Choice A makes calculation error by underestimating Δt_min, resulting in incorrect feasibility assessment. Optimization strategy: (1) calculate minimum parameter needed (Δt_min = mv/F_max or d_min = KE/F_max), (2) check available designs against this minimum, (3) select design meeting safety requirement with least excess (closest to minimum while still safe), (4) verify constraints satisfied. Designs significantly exceeding minimum waste resources; designs below minimum fail safety criteria.
A 1200 kg car hits a rigid barrier at $v=15\ \text{m/s}$ . The front-end crumple zone length $d$ can be designed to reduce average impact force, modeled by $F\approx \text{KE}/d$ , where $\text{KE}=\tfrac12 mv^2$ . Safety requires $F\le 2.0\times 10^5\ \text{N}$ . However, styling constraints limit the crumple distance to $d\le 0.80\ \text{m}$ . What is the minimum crumple distance $d_{\min}$ needed, and is it within the design limit?
Answer: $d_{\min}=0.68\ \text{m}$ ; within limit because $0.68\le 0.80$ This question tests understanding of optimizing collision safety designs using impulse-momentum or work-energy to minimize forces while meeting constraints. To minimize force for a given momentum change Δp = mv, solve F_max = Δp/Δt_min for minimum time: Δt_min = mv/F_max, or for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 1200 kg impacting at v = 15 m/s, the kinetic energy is KE = ½mv² = 0.5*1200*225 = 135000 J. To keep force below F_max = 200000 N, the minimum crumple distance is d_min = KE/F_max = 135000/200000 = 0.675 m. The styling limit allows up to d = 0.80 m which is greater than this minimum, so F_avg can be kept <= threshold. Choice B is correct because it correctly calculates d_min using the appropriate formula and properly evaluates against both safety goal and constraints. Choice C makes calculation error by overestimating d_min, perhaps forgetting the 1/2 in KE, leading to incorrect feasibility. Optimization strategy: (1) calculate minimum parameter needed (Δt_min = mv/F_max or d_min = KE/F_max), (2) check available designs against this minimum, (3) select design meeting safety requirement with least excess (closest to minimum while still safe), (4) verify constraints satisfied. Designs significantly exceeding minimum waste resources; designs below minimum fail safety criteria.
A $900\,\text{kg}$ compact car hits a barrier at $v=14\,\text{m/s}$ . You are choosing a crumple zone length $d$ to keep average force below $F_{\max}=180{,}000\,\text{N}$ . Constraint: you must keep $d\le 0.60\,\text{m}$ to meet styling and packaging. Which option is the minimum crumple distance that meets the force limit (and also satisfies the constraint)? Use $F\approx \dfrac{\tfrac12 mv^2}{d}$ .
Answer: $d=0.50\,\text{m}$ This question tests understanding of optimizing collision safety designs using work-energy to minimize forces while meeting constraints. To minimize force for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 900 kg impacting at v = 14 m/s, the kinetic energy is KE = ½(900)(14²) = ½(900)(196) = 88,200 J. To keep force below F_max = 180,000 N, the minimum crumple distance is d_min = KE/F_max = 88,200/180,000 = 0.49 m. Since we need the minimum distance that meets the force limit and the constraint d ≤ 0.60 m, we need d ≥ 0.49 m. Choice C is correct because d = 0.50 m is the smallest option that exceeds d_min = 0.49 m, thus meeting the force requirement with minimal excess. Choice A (d = 0.20 m) would give F = 88,200/0.20 = 441,000 N > 180,000 N, failing safety. Optimization strategy: (1) calculate minimum parameter needed (d_min = KE/F_max = 0.49 m), (2) check available designs against this minimum, (3) select design meeting safety requirement with least excess (0.50 m is closest to 0.49 m while still safe), (4) verify constraints satisfied (0.50 < 0.60). Designs below minimum fail safety criteria.
A $m=70\,\text{kg}$ test dummy moves at $v=14\,\text{m/s}$ and is restrained by a seatbelt pretensioner system. You can tune it to increase stopping time $\Delta t$ to reduce average force. Safety requirement: $F_{\max}=2500\,\text{N}$ . System constraint: due to belt stretch limits, $\Delta t$ cannot exceed $0.35\,\text{s}$ . What is the minimum stopping time needed, and which conclusion is correct? (Use $\Delta t_{\min}=\frac{mv}{F_{\max}}$ .)
Answer: $\Delta t_{\min}=0.39\,\text{s}$ ; not achievable because $0.39>0.35$ . This question tests understanding of optimizing collision safety designs using impulse-momentum or work-energy to minimize forces while meeting constraints. To minimize force for a given momentum change Δp = mv, solve F_max = Δp/Δt_min for minimum time: Δt_min = mv/F_max, or for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 70 kg moving at v = 14 m/s, the momentum change is Δp = mv = 70 × 14 = 980 kg⋅m/s. To keep force below F_max = 2500 N, the minimum stopping time is Δt_min = Δp/F_max = 980/2500 = 0.392 s ≈ 0.39 s. Design constraint limits Δt to 0.35 s maximum, which is less than Δt_min = 0.39 s, so F_avg would exceed 2500 N threshold. Choice A is correct because it states Δt_min = 0.39 s and correctly concludes it's not achievable since 0.39 > 0.35. Choice C incorrectly states Δt_min = 0.039 s (one-tenth the correct value). Optimization strategy: (1) calculate minimum parameter needed (Δt_min = mv/F_max or d_min = KE/F_max), (2) check available designs against this minimum, (3) select design meeting safety requirement with least excess (closest to minimum while still safe), (4) verify constraints satisfied. Designs significantly exceeding minimum waste resources; designs below minimum fail safety criteria.
A $50\,\text{kg}$ test dummy in a sled test is moving at $14\,\text{m/s}$ when it hits a restraint system. Engineers can adjust only the stopping time $\Delta t$ by changing webbing stretch. Safety goal: $F \le 2000\,\text{N}$ . Constraint: to prevent the dummy from contacting the dashboard, $\Delta t$ must be $\le 0.50\,\text{s}$ . What is the minimum $\Delta t$ required, and does it satisfy the dashboard constraint?
Answer: $\Delta t_{\min}=0.35\,\text{s}$ , satisfies ( $0.35\le 0.50$ ) This question tests understanding of optimizing collision safety designs using impulse-momentum or work-energy to minimize forces while meeting constraints. To minimize force for a given momentum change Δp = mv, solve F_max = Δp/Δt_min for minimum time: Δt_min = mv/F_max, or for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 50 kg impacting at v = 14 m/s, the momentum change is Δp = mv = 50*14 = 700 kg⋅m/s. To keep force below F_max = 2000 N, the minimum collision time is Δt_min = Δp/F_max = 700/2000 = 0.35 s. Choice B provides Δt_min=0.35 s which is less than or equal to the constraint of 0.50 s, so satisfies with F_avg ≤ threshold. Choice B is correct because it correctly calculates Δt_min using appropriate formula and properly evaluates against both safety goal and constraints. Choice C makes calculation error by overestimating Δt_min to 0.70 s, incorrectly claiming violates constraint. Optimization strategy: (1) calculate minimum parameter needed (Δt_min = mv/F_max or d_min = KE/F_max), (2) check available designs against this minimum, (3) select design meeting safety requirement with least excess (closest to minimum while still safe), (4) verify constraints satisfied. Designs significantly exceeding minimum waste resources; designs below minimum fail safety criteria.
A $1800\,\text{kg}$ SUV traveling at $18\,\text{m/s}$ must have an average collision force no greater than $F_{\max}=350{,}000\,\text{N}$ . You can choose one of four crumple-zone designs (same cost), each giving a different deformation distance $d$ . Which design is the minimum deformation distance that still meets the force requirement? Use $F \approx \dfrac{\tfrac12 m v^2}{d}$ . Options: (1) $d=0.60\,\text{m}$ , (2) $d=0.80\,\text{m}$ , (3) $d=0.90\,\text{m}$ , (4) $d=1.20\,\text{m}$ .
Answer: Design (3) $d=0.90\,\text{m}$ This question tests understanding of optimizing collision safety designs using work-energy to minimize forces while meeting constraints. To minimize force for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 1800 kg impacting at v = 18 m/s, the kinetic energy is KE = ½(1800)(18²) = 291,600 J. To keep force below F_max = 350,000 N, the minimum crumple distance is d_min = 291,600/350,000 = 0.833 m. Design (3) with d = 0.90 m provides the minimum distance that exceeds this requirement, so F_avg = 291,600/0.90 = 324,000 N < 350,000 N threshold. Choice C is correct because it identifies design (3) as the minimum deformation distance that still meets the force requirement. Choices A and B (0.60 m and 0.80 m) would result in forces of 486,000 N and 364,500 N respectively, both exceeding the limit. Optimization strategy: (1) calculate minimum parameter needed (d_min = 0.833 m), (2) check available designs against this minimum, (3) select design meeting safety requirement with least excess (0.90 m is closest to 0.833 m while still safe), (4) verify constraints satisfied. Design (4) at 1.20 m would work but wastes resources.
A 1400 kg car at 18 m/s must keep average impact force below $F_{\max}=2.5\times 10^5\ \text{N}$ using a crumple zone. The design parameter is deformation distance $d$ , but the maximum available distance is 0.80 m. If the design uses the full 0.80 m, what average force results, and does it meet the requirement? (Use $F=\text{KE}/d$ .)
Answer: $F\approx 2.8\times 10^5\ \text{N}$ ; does not meet the requirement This question tests understanding of optimizing collision safety designs using work-energy to minimize forces while meeting constraints. To minimize force for given energy KE = ½mv², solve F_max = KE/d for the actual force when using a specific distance d—we need to check if this meets the safety threshold. For this scenario with m = 1400 kg impacting at v = 18 m/s, the kinetic energy is KE = ½mv² = ½(1400)(18²) = ½(1400)(324) = 226,800 J. Using the full available crumple distance d = 0.80 m, the average force is F = KE/d = 226,800/0.80 = 283,500 N ≈ 2.84×10⁵ N. Since this exceeds F_max = 2.5×10⁵ N, the design does not meet the requirement. Choice B is correct because it correctly calculates F ≈ 2.8×10⁵ N (rounded from 2.835×10⁵ N) and identifies that this exceeds the 2.5×10⁵ N threshold. Choice A incorrectly calculates F ≈ 1.1×10⁵ N; Choices C and D give wrong force values and incorrect conclusions. Optimization strategy: (1) calculate kinetic energy KE = ½mv², (2) determine force using available distance F = KE/d, (3) compare to threshold. Since 2.84×10⁵ N > 2.5×10⁵ N, the design fails—either more crumple distance is needed (d_min = 226,800/250,000 = 0.907 m) or impact energy must be reduced.
A $1400\,\text{kg}$ sedan crashes at $v=22\,\text{m/s}$ . Engineers can choose between four crumple-zone designs (different $d$ values). The safety requirement is $F\le F_{\max}=500{,}000\,\text{N}$ . The packaging constraint is $d\le 0.75\,\text{m}$ . Optimization goal: select the design that meets both constraints while using the smallest $d$ (to save space/cost). Which $d$ is optimal? Use $F\approx \dfrac{\tfrac12 mv^2}{d}$ .
Answer: $d=0.68\,\text{m}$ This question tests understanding of optimizing collision safety designs using work-energy to minimize forces while meeting constraints. To minimize force for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 1400 kg impacting at v = 22 m/s, the kinetic energy is KE = ½(1400)(22²) = ½(1400)(484) = 338,800 J. To keep force below F_max = 500,000 N, the minimum crumple distance is d_min = KE/F_max = 338,800/500,000 = 0.6776 m ≈ 0.68 m. Since the constraint requires d ≤ 0.75 m, we need to select from the given options the smallest d that meets d ≥ 0.68 m. Choice C is correct because d = 0.68 m equals the minimum required distance, thus meeting the force requirement with no excess while satisfying the constraint (0.68 < 0.75). Choices A (0.50 m) and B (0.60 m) would result in forces exceeding 500,000 N. Optimization strategy: (1) calculate minimum parameter needed (d_min = KE/F_max = 0.68 m), (2) check available designs against this minimum, (3) select design meeting safety requirement with least excess (0.68 m exactly meets requirement), (4) verify constraints satisfied. This represents optimal design with zero waste.
A 1200 kg car traveling at $20\,\text{m/s}$ must be designed with a front-end crumple zone that limits the average collision force to $F_{\max}=300{,}000\,\text{N}$ . Assume the crumple zone provides a stopping distance $d$ and use $F \approx \dfrac{\tfrac12 m v^2}{d}$ . The vehicle design allows at most $d=0.90\,\text{m}$ of deformation. What is the minimum crumple distance required, and is the design feasible within the $0.90\,\text{m}$ limit?
Answer: $d_{\min}=0.80\,\text{m}$ ; feasible because $0.80\,\text{m}<0.90\,\text{m}$ This question tests understanding of optimizing collision safety designs using work-energy to minimize forces while meeting constraints. To minimize force for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 1200 kg impacting at v = 20 m/s, the kinetic energy is KE = ½(1200)(20²) = 240,000 J. To keep force below F_max = 300,000 N, the minimum crumple distance is d_min = 240,000/300,000 = 0.80 m. Design with d = 0.80 m provides exactly the minimum needed, so F_avg = 240,000/0.80 = 300,000 N = threshold. Choice A is correct because it correctly calculates d_min = 0.80 m using the work-energy formula and identifies that this is feasible within the 0.90 m limit. Choice B incorrectly calculates d_min = 1.60 m, which would give F = 240,000/1.60 = 150,000 N (half the threshold). Optimization strategy: (1) calculate minimum parameter needed (d_min = KE/F_max = 0.80 m), (2) check available designs against this minimum, (3) select design meeting safety requirement with least excess, (4) verify constraints satisfied. Designs significantly exceeding minimum waste resources; designs below minimum fail safety criteria.
A 90 kg stunt performer moving at 12 m/s is stopped by a safety net. You can vary the net's stopping time $\Delta t$ (by adjusting slack) to reduce force. The safety requirement is that the average force must be $\le 3000\ \text{N}$ . Due to stage constraints, the net can provide at most $0.25\ \text{s}$ of stopping time. What is the minimum stopping time needed to satisfy the force limit, and is it achievable? Use $F_{\text{avg}}=\Delta p/\Delta t$ with $\Delta p=mv$ .
Answer: $\Delta t_{\min}=0.36\ \text{s}$ , not achievable because $0.36>0.25$ This question tests understanding of optimizing collision safety designs using impulse-momentum or work-energy to minimize forces while meeting constraints. To minimize force for a given momentum change Δp = mv, solve F_max = Δp/Δt_min for minimum time: Δt_min = mv/F_max, or for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 90 kg impacting at v = 12 m/s, the momentum change is Δp = mv = 90*12 = 1080 kg⋅m/s. To keep force below F_max = 3000 N, the minimum collision time is Δt_min = Δp/F_max = 1080/3000 = 0.36 s. The available net provides Δt = 0.25 s which is less than this minimum, so F_avg = 1080/0.25 = 4320 N > threshold. Choice B is correct because it correctly calculates Δt_min using the appropriate formula and properly evaluates against both safety goal and constraints. Choice A makes calculation error by using incorrect values, selecting insufficient Δt resulting in F > threshold. Optimization strategy: (1) calculate minimum parameter needed (Δt_min = mv/F_max or d_min = KE/F_max), (2) check available designs against this minimum, (3) select design meeting safety requirement with least excess (closest to minimum while still safe), (4) verify constraints satisfied. Designs significantly exceeding minimum waste resources; designs below minimum fail safety criteria.
A 75 kg cyclist hits a padded barrier at $v=9.0\ \text{m/s}$ and is brought to rest. The barrier design parameter is the collision time $\Delta t$ (longer time lowers force). Safety requires $F_{\text{avg}}\le 1500\ \text{N}$ using $F_{\text{avg}}=\Delta p/\Delta t$ . Due to material rebound issues, the barrier cannot extend the collision longer than $\Delta t\le 0.40\ \text{s}$ . Which conclusion is correct?
Answer: Not feasible; $\Delta t_{\min}=0.45\ \text{s}$ so $\Delta t_{\min}>0.40\ \text{s}$ This question tests understanding of optimizing collision safety designs using impulse-momentum or work-energy to minimize forces while meeting constraints. To minimize force for a given momentum change Δp = mv, solve F_max = Δp/Δt_min for minimum time: Δt_min = mv/F_max, or for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 75 kg impacting at v = 9.0 m/s, the momentum change is Δp = mv = 75*9 = 675 kg⋅m/s. To keep force below F_max = 1500 N, the minimum collision time is Δt_min = Δp/F_max = 675/1500 = 0.45 s. The material limit allows up to Δt = 0.40 s which is less than this minimum, so not feasible. Choice B is correct because it correctly calculates Δt_min using the appropriate formula and properly evaluates that it is not feasible within the constraint. Choice A makes calculation error by underestimating Δt_min, incorrectly concluding feasibility. Optimization strategy: (1) calculate minimum parameter needed (Δt_min = mv/F_max or d_min = KE/F_max), (2) check available designs against this minimum, (3) select design meeting safety requirement with least excess (closest to minimum while still safe), (4) verify constraints satisfied. Designs significantly exceeding minimum waste resources; designs below minimum fail safety criteria.
A 60 kg skier moving at 12 m/s crashes into a safety net. To minimize injury, the average impact force must be kept below $F_{\max}=1800\ \text{N}$ . The design parameter you can vary is the net's stopping time $\Delta t$ (by adjusting net slack). The net cannot allow a stopping time longer than 0.60 s due to space limits behind it. What is the minimum stopping time required to keep the average force below the limit, and is it feasible within the 0.60 s constraint? (Use $F_{\text{avg}}=\Delta p/\Delta t$ .)
Answer: $\Delta t_{\min}=0.40\ \text{s}$ ; feasible because $0.40\le 0.60$ This question tests understanding of optimizing collision safety designs using impulse-momentum to minimize forces while meeting constraints. To minimize force for a given momentum change Δp = mv, solve F_max = Δp/Δt_min for minimum time: Δt_min = mv/F_max—designs must provide at least this minimum to keep forces below safety thresholds. For this scenario with m = 60 kg impacting at v = 12 m/s, the momentum change is Δp = mv = 60 × 12 = 720 kg⋅m/s. To keep force below F_max = 1800 N, the minimum collision time is Δt_min = Δp/F_max = 720/1800 = 0.40 s. Since the net allows up to 0.60 s, this design is feasible because 0.40 ≤ 0.60, so F_avg = 1800 N ≤ threshold. Choice B is correct because it correctly calculates Δt_min = 0.40 s using the impulse-momentum formula and identifies that this meets the safety threshold within the 0.60 s constraint. Choice A incorrectly calculates Δt_min = 0.25 s which would result in F = 720/0.25 = 2880 N > 1800 N threshold; Choice C gives 0.60 s which exceeds the minimum needed; Choice D gives 0.80 s which violates the 0.60 s constraint. Optimization strategy: (1) calculate minimum parameter needed (Δt_min = mv/F_max), (2) check against maximum allowed time, (3) select design meeting safety requirement with least excess. Designs significantly exceeding minimum waste resources; designs below minimum fail safety criteria.
A 1000 kg car traveling at 25 m/s needs a redesigned front end. The design parameter is crumple distance $d$ . Safety requires $F\le 3.0\times 10^5\ \text{N}$ , but styling constraints cap the crumple zone at $d_{\max}=0.90\ \text{m}$ . Determine whether the force limit can be met, and if not, which change would make it feasible while keeping the same force limit. (Use $F=\text{KE}/d$ .)
Answer: Not feasible; required $d_{\min}=1.04\ \text{m}$ . To make it feasible, increase $d_{\max}$ above 1.04 m (or reduce speed/mass). This question tests understanding of optimizing collision safety designs using work-energy to minimize forces while meeting constraints. To minimize force for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = KE/F_max—designs must provide at least this minimum to keep forces below safety thresholds. For this scenario with m = 1000 kg impacting at v = 25 m/s, the kinetic energy is KE = ½mv² = ½(1000)(25²) = ½(1000)(625) = 312,500 J. To keep force below F_max = 3.0×10⁵ N, the minimum crumple distance is d_min = KE/F_max = 312,500/300,000 = 1.042 m ≈ 1.04 m. Since the styling constraint limits d_max to 0.90 m, this requirement is not feasible because 1.04 > 0.90, meaning the force would exceed the safety limit. Choice B is correct because it correctly calculates d_min = 1.04 m and identifies that this exceeds the 0.90 m constraint, making the current design infeasible, and suggests increasing d_max above 1.04 m or reducing speed/mass. Choice A incorrectly states d_min = 0.52 m; Choice C gives wrong calculation; Choice D incorrectly claims 1.04 < 0.90. Optimization strategy: (1) calculate minimum parameter needed (d_min = KE/F_max), (2) check against constraint, (3) if minimum exceeds maximum allowed, identify needed changes. To make feasible while keeping same force limit, either increase allowed crumple distance beyond 1.04 m or reduce impact energy through lower speed or mass.
A $1000\,\text{kg}$ car at $25\,\text{m/s}$ needs a redesigned crumple zone. The goal is to keep the average force below $F_{\max}=400{,}000\,\text{N}$ by increasing the deformation distance $d$ . The engineering team proposes $d=0.60\,\text{m}$ , but the maximum possible is $0.80\,\text{m}$ . Using $F \approx \dfrac{\tfrac12 m v^2}{d}$ , which statement is correct?
Answer: $d_{\min}=0.78\,\text{m}$ , so $0.60\,\text{m}$ is unsafe but $0.80\,\text{m}$ would be safe This question tests understanding of optimizing collision safety designs using work-energy to minimize forces while meeting constraints. To minimize force for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 1000 kg impacting at v = 25 m/s, the kinetic energy is KE = ½(1000)(25²) = 312,500 J. To keep force below F_max = 400,000 N, the minimum crumple distance is d_min = 312,500/400,000 = 0.78125 m ≈ 0.78 m. Design with d = 0.60 m would result in F_avg = 312,500/0.60 ≈ 520,833 N > 400,000 N threshold, making it unsafe. Choice A is correct because it correctly calculates d_min = 0.78 m and identifies that 0.60 m is unsafe while 0.80 m would be safe (giving F = 312,500/0.80 = 390,625 N < 400,000 N). Choice C incorrectly calculates d_min = 1.56 m, which would give F = 312,500/1.56 ≈ 200,321 N (half the threshold). Optimization strategy: (1) calculate minimum parameter needed (d_min = KE/F_max = 0.78 m), (2) check proposed designs against this minimum, (3) select design meeting safety requirement with least excess. The 0.80 m maximum design barely meets the requirement.
A $70\,\text{kg}$ passenger in a car experiences a sudden stop in a crash. Just before impact the passenger's speed is $15\,\text{m/s}$ . You are designing an airbag that increases the stopping time to reduce the average force on the passenger. The safety requirement is $F_{\max}=3000\,\text{N}$ (average force), and the airbag system has a packaging constraint that the stopping time it can provide is at most $0.40\,\text{s}$ . Using $F_{\text{avg}}=\Delta p/\Delta t$ , what is the minimum stopping time $\Delta t_{\min}$ required to keep the average force at or below $3000\,\text{N}$ , and is it feasible under the $0.40\,\text{s}$ limit?
Answer: $\Delta t_{\min}=0.35\,\text{s}$ ; feasible This question tests understanding of optimizing collision safety designs using impulse-momentum or work-energy to minimize forces while meeting constraints. To minimize force for a given momentum change Δp = mv, solve F_max = Δp/Δt_min for minimum time: Δt_min = mv/F_max, or for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 70 kg impacting at v = 15 m/s, the momentum change is Δp = mv = 70 × 15 = 1050 kg⋅m/s. To keep force below F_max = 3000 N, the minimum collision time is Δt_min = Δp/F_max = 1050/3000 = 0.35 s. Since the airbag constraint allows up to 0.40 s, this design is feasible as 0.35 s < 0.40 s. Choice B is correct because it correctly calculates Δt_min = 0.35 s using the impulse-momentum formula and identifies that this meets the safety threshold within the packaging constraint. Choice A incorrectly calculates 0.035 s (off by factor of 10), Choice C gives 0.50 s which exceeds the required minimum, and Choice D gives 0.23 s which is less than required and would result in F > 3000 N. Optimization strategy: (1) calculate minimum parameter needed (Δt_min = mv/F_max = 1050/3000 = 0.35 s), (2) check against constraint (0.35 s < 0.40 s limit), (3) select design meeting safety requirement with least excess. Designs significantly exceeding minimum waste resources; designs below minimum fail safety criteria.
A $1500\,\text{kg}$ vehicle traveling at $25\,\text{m/s}$ must be redesigned with a longer crumple zone to reduce average force. The target is $F_{\max}=3.0\times 10^5\,\text{N}$ . The only design parameter is crumple distance $d$ . Constraint: the vehicle architecture allows at most $0.90\,\text{m}$ of controlled deformation. Determine whether meeting the force limit is feasible, using $F=\text{KE}/d$ .
Answer: Not feasible, because $d_{\min}=1.56\,\text{m} > 0.90\,\text{m}$ This question tests understanding of optimizing collision safety designs using impulse-momentum or work-energy to minimize forces while meeting constraints. To minimize force for a given momentum change Δp = mv, solve F_max = Δp/Δt_min for minimum time: Δt_min = mv/F_max, or for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 1500 kg impacting at v = 25 m/s, the kinetic energy is KE = ½mv² = 0.5*1500*625 = 468750 J. To keep force below F_max = 300000 N, the minimum distance is d_min = KE/F_max = 468750/300000 = 1.5625 m. Choice B identifies d_min ≈1.56 m which exceeds the constraint of 0.90 m, so not feasible with F_avg > threshold if d=0.90 m. Choice B is correct because it correctly calculates d_min using appropriate formula and properly evaluates against both safety goal and constraints. Choice A makes calculation error by underestimating d_min to 0.78 m, incorrectly claiming feasible. Optimization strategy: (1) calculate minimum parameter needed (Δt_min = mv/F_max or d_min = KE/F_max), (2) check available designs against this minimum, (3) select design meeting safety requirement with least excess (closest to minimum while still safe), (4) verify constraints satisfied. Designs significantly exceeding minimum waste resources; designs below minimum fail safety criteria.
A $90\,\text{kg}$ rider on an e-scooter hits a wall at $v=10\,\text{m/s}$ . An airbag can increase the collision time $\Delta t$ . Optimization goal: keep average force below $F_{\max}=3000\,\text{N}$ by choosing the minimum required collision time. Constraint: due to sensor and bag limits, the system can provide at most $\Delta t=0.20\,\text{s}$ . Using $F_{\text{avg}}=\dfrac{\Delta p}{\Delta t}$ with $\Delta p\approx mv$ , what is $\Delta t_{\min}$ and is it achievable?
Answer: $\Delta t_{\min}=0.30\,\text{s}$ ; not achievable because $0.30>0.20$ This question tests understanding of optimizing collision safety designs using impulse-momentum to minimize forces while meeting constraints. To minimize force for a given momentum change Δp = mv, solve F_max = Δp/Δt_min for minimum time: Δt_min = mv/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 90 kg impacting at v = 10 m/s, the momentum change is Δp = mv = (90)(10) = 900 kg⋅m/s. To keep force below F_max = 3000 N, the minimum collision time is Δt_min = Δp/F_max = 900/3000 = 0.30 s. The constraint allows at most Δt = 0.20 s, which is less than the required 0.30 s, so the design is not achievable. Choice A is correct because it correctly calculates Δt_min = 0.30 s and identifies that this exceeds the 0.20 s limit, making it not achievable. Choice B incorrectly calculates Δt_min as 0.03 s, which would give F = 900/0.03 = 30,000 N >> 3000 N. Optimization strategy: (1) calculate minimum parameter needed (Δt_min = mv/F_max), (2) check available designs against this minimum, (3) verify constraints satisfied. Designs below minimum fail safety criteria; this system cannot provide sufficient collision time.
A $90\,\text{kg}$ test dummy moves at $12\,\text{m/s}$ just before being restrained by a safety harness. You can adjust the harness "give" (stopping time) by changing webbing and load limiter settings. The goal is to minimize average force on the dummy while meeting the requirement $F\le 2500\,\text{N}$ . A design constraint is that the harness system cannot allow a stopping time longer than $0.50\,\text{s}$ (to prevent excessive forward motion). Using $F_{\text{avg}}=\Delta p/\Delta t$ , what is the minimum stopping time required, and does it violate the $0.50\,\text{s}$ constraint?
Answer: $\Delta t_{\min}=0.43\,\text{s}$ ; does not violate This question tests understanding of optimizing collision safety designs using impulse-momentum or work-energy to minimize forces while meeting constraints. To minimize force for a given momentum change Δp = mv, solve F_max = Δp/Δt_min for minimum time: Δt_min = mv/F_max, or for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 90 kg impacting at v = 12 m/s, the momentum change is Δp = mv = 90 × 12 = 1080 kg⋅m/s. To keep force below F_max = 2500 N, the minimum collision time is Δt_min = Δp/F_max = 1080/2500 = 0.432 s ≈ 0.43 s. Since the constraint limits stopping time to 0.50 s maximum, this design does not violate the constraint as 0.43 s < 0.50 s. Choice A is correct because it correctly calculates Δt_min = 0.43 s and identifies that this does not violate the 0.50 s constraint. Choice B incorrectly calculates 0.22 s, Choice C gives 0.50 s which is higher than needed, and Choice D gives 0.86 s which both exceeds the requirement and violates the constraint. Optimization strategy: (1) calculate minimum parameter needed (Δt_min = 1080/2500 = 0.43 s), (2) check against constraint (0.43 s < 0.50 s limit), (3) select design meeting safety requirement with least excess. The harness can be set to 0.43 s to minimize force while staying within limits.
A $m=60\,\text{kg}$ skier hits a safety net at $v=10\,\text{m/s}$ . The net system is designed to increase collision time to reduce average force. Safety requirement: $F_{\max}=1500\,\text{N}$ . Due to installation constraints, the longest achievable stopping time is $\Delta t_{\text{max}}=0.30\,\text{s}$ . What is the minimum stopping time required, and what conclusion follows? (Use $\Delta t_{\min}=\frac{mv}{F_{\max}}$ .)
Answer: $\Delta t_{\min}=0.40\,\text{s}$ ; not feasible because $0.40>0.30$ . This question tests understanding of optimizing collision safety designs using impulse-momentum or work-energy to minimize forces while meeting constraints. To minimize force for a given momentum change Δp = mv, solve F_max = Δp/Δt_min for minimum time: Δt_min = mv/F_max, or for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 60 kg moving at v = 10 m/s, the momentum change is Δp = mv = 60 × 10 = 600 kg⋅m/s. To keep force below F_max = 1500 N, the minimum stopping time is Δt_min = Δp/F_max = 600/1500 = 0.40 s. Design constraint allows Δt_max = 0.30 s, which is less than Δt_min = 0.40 s, so F_avg would exceed 1500 N threshold. Choice A is correct because it states Δt_min = 0.40 s and correctly concludes it's not feasible since 0.40 > 0.30. Choice B incorrectly states Δt_min = 0.20 s (half the correct value). Optimization strategy: (1) calculate minimum parameter needed (Δt_min = mv/F_max or d_min = KE/F_max), (2) check available designs against this minimum, (3) select design meeting safety requirement with least excess (closest to minimum while still safe), (4) verify constraints satisfied. Designs significantly exceeding minimum waste resources; designs below minimum fail safety criteria.
A $1000\,\text{kg}$ car traveling at $22\,\text{m/s}$ is being redesigned. Engineers can choose either to (i) increase crumple distance $d$ or (ii) accept the current $d=0.70\,\text{m}$ . The safety requirement is $F \le 4.0\times 10^5\,\text{N}$ using $F=\text{KE}/d$ . Constraint: due to styling, $d$ cannot exceed $0.80\,\text{m}$ . What minimum $d$ is required, and what design conclusion follows?
Answer: $d_{\min}=0.61\,\text{m}$ ; current $0.70\,\text{m}$ already meets the force limit This question tests understanding of optimizing collision safety designs using impulse-momentum or work-energy to minimize forces while meeting constraints. To minimize force for a given momentum change Δp = mv, solve F_max = Δp/Δt_min for minimum time: Δt_min = mv/F_max, or for given energy KE = ½mv², solve F_max = KE/d_min for minimum distance: d_min = (½mv²)/F_max—designs must provide at least these minimums to keep forces below safety thresholds. For this scenario with m = 1000 kg impacting at v = 22 m/s, the kinetic energy is KE = ½mv² = 0.5*1000*484 = 242000 J. To keep force below F_max = 400000 N, the minimum distance is d_min = KE/F_max = 242000/400000 ≈ 0.605 m. The current design provides d = 0.70 m which is greater than this minimum, so F_avg ≈ 345714 N < threshold. Choice A is correct because it correctly calculates d_min using appropriate formula and identifies that current design meets safety threshold within constraints. Choice B makes calculation error by overestimating d_min to 1.21 m, incorrectly claiming not possible. Optimization strategy: (1) calculate minimum parameter needed (Δt_min = mv/F_max or d_min = KE/F_max), (2) check available designs against this minimum, (3) select design meeting safety requirement with least excess (closest to minimum while still safe), (4) verify constraints satisfied. Designs significantly exceeding minimum waste resources; designs below minimum fail safety criteria.

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