A 54.5-kg passenger in the back seat of a van
is wearing a seat belt when the van runs into
a stone wall. The van is originally moving at
12.6 m/s. The front end of the van collapses
0.624 m as the van comes to rest.
(a) Determine the passenger’s kinetic energy
before the crash.
(b) Find the average force exerted on the
passenger by the seat belt.
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To answer these questions, we will need to use the principles of kinetic energy and work. Here's how you can find the answers:
(a) To determine the passenger's kinetic energy before the crash, we can use the formula for kinetic energy:
Kinetic energy (KE) = 1/2 * mass * velocity^2
Given:
- Mass of the passenger (m) = 54.5 kg
- Initial velocity of the van (v) = 12.6 m/s
Using the formula, we can calculate the kinetic energy:
KE = 1/2 * 54.5 kg * (12.6 m/s)^2
= 1/2 * 54.5 kg * 158.76 m^2/s^2
= 1/2 * 54.5 kg * 158.76 J
= 4255.46 J
Therefore, the passenger's kinetic energy before the crash is 4255.46 Joules.
(b) To find the average force exerted on the passenger by the seat belt, we can use the work-energy principle. The work done on the passenger is equal to the change in kinetic energy:
Work (W) = Change in KE
Given:
- Displacement of the front end of the van (d) = 0.624 m
Using the work-energy principle, we have:
W = Change in KE = KE_final - KE_initial
Since the van comes to rest, the passenger's final kinetic energy is zero:
KE_final = 0 J
Therefore, we can write the equation:
W = 0 J - 4255.46 J
W = - 4255.46 J
The work done on the passenger is negative because it is done by an external force (the seat belt) against the direction of the motion. The seat belt exerts a restraining force on the passenger during the collision.
Now, we can calculate the average force:
Average force (F) = Work / Displacement
F = (- 4255.46 J) / 0.624 m
F ≈ - 6818.97 N
Therefore, the average force exerted on the passenger by the seat belt is approximately 6818.97 Newtons (N). The negative sign indicates that the force is in the opposite direction of the displacement.
Note: Negative values indicate the direction opposite to the initial motion.