Question 6.1: How Good Are the Bumpers? Goal Find an impulse and estimate ...
How Good Are the Bumpers? Goal Find an impulse and estimate a force in a collision of a moving object with a stationary object.
Problem In a crash test, a car of mass 1.50 × 10³ kg collides with a wall and rebounds as in Figure 6.3. The initial and final velocities of the car are v_{i}=-15.0 m / s and v_{f}=2.60 m / s , respectively. If the collision lasts for 0.150 s, find (a) the impulse delivered to the car due to the collision and (b) the size and direction of the average force exerted on the car.
Strategy Here the initial and final momenta are both nonzero. Find the momenta and substitute into the impulse–momentum theorem, Equation 6.6, solving for F_{\text {av }} .
\overrightarrow{ F }_{ av } \Delta t=\Delta \overrightarrow{ p }
(a) Find the impulse delivered to the car.
Calculate the initial and final momenta of the car:
\begin{aligned}p_{i} &=m v_{i}=\left(1.50 \times 10^{3} kg \right)(-15.0 m / s ) \\&=-2.25 \times 10^{4} kg \cdot m / s \\p_{f} &=m v_{f}=\left(1.50 \times 10^{3} kg \right)(+2.60 m / s ) \\&=+0.390 \times 10^{4} kg \cdot m / s\end{aligned}
The impulse is just the difference between the final and initial momenta:
\begin{aligned}I &=p_{f}-p_{i} \\&=+0.390 \times 10^{4} kg \cdot m / s -\left(-2.25 \times 10^{4} kg \cdot m / s \right) \\I &=2.64 \times 10^{4} kg \cdot m / s\end{aligned}
(b) Find the average force exerted on the car.
Apply Equation 6.6, the impulse–momentum theorem:
F_{ av }=\frac{\Delta p}{\Delta t}=\frac{2.64 \times 10^{4} kg \cdot m / s }{0.150 s }=+1.76 \times 10^{5} N
Remarks When the car doesn’t rebound off the wall, the average force exerted on the car is smaller than the value just calculated. With a final momentum of zero, the car undergoes a smaller change in momentum.