Optimize Designs for Collision Safety
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Physics › Optimize Designs for Collision Safety
A 60 kg test dummy moves at $v=12\ \text{m/s}$ before hitting a padded wall and stopping. Designers can tune the padding to increase the collision time $\Delta t$. Safety requires the average force to be at most $F_{\max}=1800\ \text{N}$ using $F_{\text{avg}}=\Delta p/\Delta t$. A mechanical constraint limits the maximum achievable collision time to $\Delta t\le 0.50\ \text{s}$. What is the minimum collision time required, and is it achievable?
$\Delta t_{\min}=0.06\ \text{s}$; not achievable because $0.06>0.50$
$\Delta t_{\min}=0.67\ \text{s}$; not achievable because $0.67>0.50$
$\Delta t_{\min}=0.40\ \text{s}$; achievable because $0.40\le 0.50$
$\Delta t_{\min}=0.25\ \text{s}$; achievable because $0.25\le 0.50$
Explanation
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 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 = 0.4 s. The mechanical limit allows up to Δt = 0.50 s which is greater than this minimum, so achievable with F_avg <= threshold. Choice A is correct because it correctly calculates Δt_min using the appropriate formula and properly evaluates against both safety goal and constraints. Choice C makes calculation error by overestimating Δt_min, incorrectly concluding not achievable. 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 kg car is tested in a crash at $v=18\ \text{m/s}$. Engineers can choose between two crumple-zone designs (single parameter is deformation distance $d$): Design A allows $d=0.50\ \text{m}$, Design B allows $d=0.80\ \text{m}$. The force limit is $F_{\max}=2.5\times 10^5\ \text{N}$ and $F\approx \text{KE}/d$. A packaging constraint requires $d\le 0.85\ \text{m}$. Which design best minimizes force while meeting all constraints?
Neither design meets the force limit
Design A, because Design B violates the packaging constraint
Design A, because smaller $d$ always reduces force
Design B, because it produces a lower force and still satisfies $d\le 0.85\ \text{m}$
Explanation
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 = 18 m/s, the kinetic energy is KE = ½mv² = 0.51000324 = 162000 J. To keep force below F_max = 250000 N, the minimum distance is d_min = KE/F_max = 162000/250000 = 0.648 m. Design A provides d=0.50 m which is less than this minimum, so F_avg = 162000/0.50 = 324000 N > threshold; Design B provides d=0.80 m which is greater than minimum, so F_avg = 162000/0.80 = 202500 N < threshold. Choice C is correct because it identifies Design B as meeting the safety threshold with lower force while satisfying the packaging constraint. Choice A is wrong because smaller d increases force, using wrong assumption. 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?
Not feasible; $\Delta t_{\min}=0.45\ \text{s}$ so $\Delta t_{\min}>0.40\ \text{s}$
Feasible; $\Delta t_{\min}=0.23\ \text{s}$ so $\Delta t_{\min}\le 0.40\ \text{s}$
Feasible; $\Delta t_{\min}=0.05\ \text{s}$ so $\Delta t_{\min}\le 0.40\ \text{s}$
Not feasible; $\Delta t_{\min}=0.10\ \text{s}$ so $\Delta t_{\min}>0.40\ \text{s}$
Explanation
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 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?
$d_{\min}=0.11\ \text{m}$; within limit because $0.11\le 0.80$
$d_{\min}=0.34\ \text{m}$; within limit because $0.34\le 0.80$
$d_{\min}=1.35\ \text{m}$; not within limit because $1.35>0.80$
$d_{\min}=0.68\ \text{m}$; within limit because $0.68\le 0.80$
Explanation
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 = 0.51200225 = 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 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 $F_{\text{avg}}=\Delta p/\Delta t$.)
$\Delta t_{\min}=0.40\ \text{s}$; feasible because $0.40\le 0.60$
$\Delta t_{\min}=0.20\ \text{s}$; feasible because $0.20\le 0.60$
$\Delta t_{\min}=0.80\ \text{s}$; not feasible because $0.80>0.60$
$\Delta t_{\min}=0.05\ \text{s}$; feasible because $0.05\le 0.60$
Explanation
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 90 kg rider on an e-scooter hits a safety barrier at $v=10\ \text{m/s}$. A deformable barrier section provides a stopping distance $d$. To minimize injury, the average force must satisfy $F\le 3000\ \text{N}$ using $F\approx \text{KE}/d$. Due to sidewalk width, the barrier can deform at most $d\le 0.50\ \text{m}$. Is it possible to meet the force limit within the deformation constraint, and what required minimum distance supports your conclusion?
Yes; $d_{\min}=0.30\ \text{m}$, which is $\le 0.50\ \text{m}$
No; $d_{\min}=1.50\ \text{m}$, which is $>0.50\ \text{m}$
No; $d_{\min}=0.05\ \text{m}$, which is $>0.50\ \text{m}$
Yes; $d_{\min}=0.15\ \text{m}$, which is $\le 0.50\ \text{m}$
Explanation
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 = 10 m/s, the kinetic energy is KE = ½mv² = 0.590100 = 4500 J. To keep force below F_max = 3000 N, the minimum deformation distance is d_min = KE/F_max = 4500/3000 = 1.5 m. The width limit allows up to d = 0.50 m which is less than this minimum, so F_avg would > threshold. Choice C is correct because it correctly calculates d_min using the appropriate formula and properly evaluates that it is not possible within the constraint. Choice A makes calculation error by underestimating d_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 1500 kg car traveling at $v=25 \text{m/s}$ must be designed so that the average collision force during a crash does not exceed $F_{\max}=3.0\times 10^5 \text{N}$. Engineers can choose the crumple zone length $d$ (single design parameter). The vehicle layout limits $d \le 1.0 \text{m}$. Using $F \approx \text{KE}/d$, which statement is correct about feasibility?
Not feasible; $d_{\min}=0.10 \text{m}$ so the design cannot meet $d \le 1.0 \text{m}$
Not feasible; $d_{\min}=1.56 \text{m}$ so the design cannot meet $d \le 1.0 \text{m}$
Feasible; $d_{\min}=0.78 \text{m}$ so the design can meet $d \le 1.0 \text{m}$
Feasible; $d_{\min}=0.26 \text{m}$ so the design can meet $d \le 1.0 \text{m}$
Explanation
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 $KE = \frac{1}{2}mv^2$, solve $F_{\max} = KE/d_{\min}$ for minimum distance: $d_{\min} = (\frac{1}{2}mv^2)/F_{\max}$—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 $KE = \frac{1}{2}mv^2 = 0.51500625 = 468750$ J. To keep force below $F_{\max} = 300000$ N, the minimum crumple distance is $d_{\min} = KE/F_{\max} = 468750/300000 = 1.5625$ m. The layout limit allows up to $d = 1.0$ m which is less than this minimum, so not feasible. Choice B is correct because it correctly calculates $d_{\min}$ using the appropriate formula and properly evaluates against both safety goal and constraints. Choice A makes calculation error by underestimating $d_{\min}$, 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 1200 kg car traveling at 18 m/s crashes head-on and is brought to rest by its crumple zone length $d$. The design goal is to minimize the average force on occupants, with a safety limit of $F_{\max}=90{,}000\ \text{N}$. The vehicle design allows at most $0.80\ \text{m}$ of front-end deformation. Assuming constant average force, what is the minimum crumple distance needed to keep $F\le F_{\max}$, and does it fit within the $0.80\ \text{m}$ limit? Use $F=\text{KE}/d$.
$d_{\min}=0.80\ \text{m}$, fits exactly within $0.80\ \text{m}$
$d_{\min}=2.16\ \text{m}$, does not fit within $0.80\ \text{m}$
$d_{\min}=1.20\ \text{m}$, does not fit within $0.80\ \text{m}$
$d_{\min}=0.22\ \text{m}$, fits within $0.80\ \text{m}$
Explanation
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 = 18 m/s, the kinetic energy is KE = ½120018² = 194400 J. To keep force below F_max = 90000 N, the minimum collision distance is d_min = 194400/90000 = 2.16 m. The available space provides d = 0.80 m which is less than this minimum, so F_avg = 194400/0.80 = 243000 N > 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 A makes calculation error by using incorrect values or formula, resulting in insufficient d_min and wrong 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 $50,\text{kg}$ 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?
$\Delta t_{\min}=0.14,\text{s}$, satisfies ($0.14\le 0.50$)
$\Delta t_{\min}=0.07,\text{s}$, satisfies
$\Delta t_{\min}=0.35,\text{s}$, satisfies ($0.35\le 0.50$)
$\Delta t_{\min}=0.70,\text{s}$, violates ($0.70>0.50$)
Explanation
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 $1000,\text{kg}$ 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?
$d_{\min}=0.61,\text{m}$; current $0.70,\text{m}$ already meets the force limit
$d_{\min}=0.80,\text{m}$; must increase to exactly $0.80,\text{m}$ to be safe
$d_{\min}=0.30,\text{m}$; current design is excessive and unsafe
$d_{\min}=1.21,\text{m}$; not possible within the $0.80,\text{m}$ limit
Explanation
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 = 0.51000484 = 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.