Practice Test 8

A fragile instrument (mass $m=10\ \text{kg}$) is shipped in a box and may be dropped from $h=1.0\ \text{m}$. The instrument can tolerate a maximum average impact force of $F_{\max}=500\ \text{N}$. Assume the cushioning compresses a distance $d$ while bringing the instrument to rest and that all gravitational potential energy becomes work done by the average stopping force: $mgh \approx F_{\text{avg}} d$.

Cushioning designs:

• Design A: bubble wrap, $d=0.05\ \text{m}$.
• Design B: foam blocks, $d=0.20\ \text{m}$.
• Design C: air pillows, $d=0.10\ \text{m}$.
• Design D: thin cardboard inserts, $d=0.02\ \text{m}$.

Which design keeps $F_{\text{avg}}<500\ \text{N}$? (Use $g\approx 9.8\ \text{m/s}^2$.)

A car at $v=25\ \text{m/s}$ must stop with an average deceleration magnitude less than $a_{\max}=30\ \text{m/s}^2$ to reduce injury risk. Assume constant deceleration and use $v^2=2ad$ to estimate the minimum stopping distance required.

Front-end designs (maximum controlled deformation distance available):

• Design A: $d=8\ \text{m}$ (too long for the vehicle), cost $\$1200.
• Design B: $d=12\ \text{m}$ (impossible for the vehicle), cost $\$1600.
• Design C: $d=10.5\ \text{m}$ (fits exactly), cost $\$1500.
• Design D: $d=6\ \text{m}$ (fits), cost $\$900.

Vehicle constraint: $d\le 10.5\ \text{m}$. Which design meets both the deceleration limit and the vehicle constraint?