At a stoplight, a large truck (3000 kg) collides with a motionless small car (1200 kg). The truck comes to an instantaneous stop; the car slides straight ahead, coming to a stop after sliding 10 meters. The measured coefficient of friction between the car’s tires and the road was 0.62. How fast was the truck moving at the moment of impact?
Strategy
At first it may seem we don’t have enough information to solve this problem. Although we know the initial speed of the car, we don’t know the speed of the truck (indeed, that’s what we’re asked to find), so we don’t know the initial momentum of the system. Similarly, we know the final speed of the truck, but not the speed of the car immediately after impact. The fact that the car eventually slid to a speed of zero doesn’t help with the final momentum, since an external friction force caused that. Nor can we calculate an impulse, since we don’t know the collision time, or the amount of time the car slid before stopping. A useful strategy is to impose a restriction on the analysis.
Suppose we define a system consisting of just the truck and the car. The momentum of this system isn’t conserved, because of the friction between the car and the road. But if we
could find the speed of the car the instant after impact—before friction had any measurable effect on the car—then we could consider the momentum of the system to be conserved, with that restriction.
Can we find the final speed of the car? Yes; we invoke the work-kinetic energy theorem.
Solution
First, define some variables. Let:
be the masses of the car and truck, respectively
be the velocities of the truck before and after the collision, respectively
Z be the velocities of the car before and after the collision, respectively
be the kinetic energies of the car immediately after the collision, and after the car has stopped sliding (so
).
d be the distance the car slides after the collision before eventually coming to a stop.
Since we actually want the initial speed of the truck, and since the truck is not part of the work-energy calculation, let’s start with conservation of momentum. For the car + truck system, conservation of momentum reads
Since the car’s initial velocity was zero, as was the truck’s final velocity, this simplifies to
So now we need the car’s speed immediately after impact. Recall that
where
Also,
The work is done over the distance the car slides, which we’ve called
d . Equating:
Friction is the force on the car that does the work to stop the sliding. With a level road, the friction force is
Since the angle between the directions of the friction force vector and the displacement
d is
, and
we have
(Notice that the car’s mass divides out; evidently the mass of the car doesn’t matter.)
Solving for the car’s speed immediately after the collision gives
