Calculate the kinetic energy, momentum, and force of a collision between a car and a deer

Question

I recently struck a deer head on at 65mph in an approximately 4,500-lb vehicle (including the weight of myself, the fuel, and other contents). The collision produced a dent in the front of the car with a maximum depth of approximately 9 inches, and I estimate the deer's weight to be about 150 lbs. In this case, the collision was almost perfectly head on, with the deer having been not moving in the direction of motion of the car prior to the collision, and thrust almost directly and horizontally in front of the car, not at an angle.

Would the kinetic energy of the car divided by the maximum depth of the dent caused by the deer give an approximation of the force of the collision? I recall reading somewhere that this is how the energy of crash tests is measured, although the depth of the dent depends on the strength of the materials. Or would this problem be more effectively modeled in terms of change in momentum? (mass of car+deer - mass of car)(velocity of car)/time the car was in contact with the deer

Note that if I were doing this problem, I would convert to SI units.

Answer 1

A few years ago, I hit a raccoon with my car the night before I wrote a kinematics exam, and so I put together a problem like this for my students. Fellow educators, beware: several students were vocally disturbed that I had reduced the death of this innocent animal to a math problem.

In that callous spirit, the lowest-assumption formulation is that the collision is “completely inelastic.” A completely inelastic collision has the maximum possible reduction in kinetic energy, because there exists a reference frame (the “center of momentum” frame) in which the final kinetic energy is zero.

In your case you know the initial momentum of your vehicle, and the initial momentum of the deer (probably zero). The “collision” approximation is that external forces are irrelevant, so that the internal momentum is conserved. That means you also know the final momentum of the (vehicle + deer) system. After a completely inelastic collision, your vehicle’s speed should have been reduced by a factor of $ \frac{m_\text{vehicle}}{m_\text{vehicle}+m_\text{deer}} = (1+\frac{m_\text{deer}}{m_\text{vehicle}})^{-1} ≈0.97 $.

The depth of the dent gives you a measure of the duration of the collision. You know that there was no dent at the onset of the collision, and that the dent was completely formed at the end of the collision. A reasonable ballpark is that your nine-inch dent was formed while your vehicle traveled eighteen inches. Since you know the speed of the vehicle, this lets you convert the impulse $\Delta p=F\Delta t$ into an average force.

If you were already braking when you actually struck the deer, consider that your initial speed may have been substantially less than the highway cruising speed.