Question

question when a person is involved in a car accident, why is the likelihood of injury greater in a head - on collision as opposed to being hit from behind? answer using the concepts of relative velocity, momentum, and average force. (select all that apply.) the change in momentum is greater in the head - on collision. the average force on the driver is greater in the head - on collision. the collapse of the crumple zone in the front of the car occurs only in the head - on collision. the momentum of the driver relative to the ground is greater in a head - on collision. the velocity of the driver relative to the ground is greater in a head - on collision.

Explanation:

Step1: Analyze change in momentum

In a head - on collision, if two cars are moving towards each other, the relative velocity of the collision is higher. Using the impulse - momentum theorem \(J = F_{avg}\Delta t=\Delta p\). If we assume similar collision times \(\Delta t\) (due to crumple zones etc.), a higher relative velocity change (for the driver) implies a larger \(\Delta p\). But if the initial velocity of the car (and thus driver) before rear - end and head - on is same (say \(v_{0}\)), in a rear - end, the final velocity \(v_{f}\) (after collision) is higher (car is pushed forward), and in a head - on \(v_{f}\) is lower (car is stopped or reversed). The change in momentum \(\Delta p=m(v_{f}-v_{0})\). For a head - on, if \(v_{f}\) is in the opposite direction (\(v_{f}=-v\), \(v>0\)), \(\Delta p=m(-v - v_{0})\), and for rear - end \(\Delta p=m(v_{f}-v_{0})\) (\(v_{f}>v_{0}\)). But if we consider relative to the other car (e.g., in head - on, if two cars of mass \(m\) moving at \(v\) towards each other, for a driver in one car, the relative velocity of the other car is \(2v\), in rear - end if a car behind is moving at \(v\) and the front car at \(u\) (\(v > u\)), relative velocity is \(v - u\)). So the change in momentum can be greater in head - on.

Step2: Analyze average force

From \(F_{avg}=\frac{\Delta p}{\Delta t}\). If \(\Delta t\) (time of collision, determined by crumple zone etc., assume similar \(\Delta t\) for both cases as crumple zones exist in front and back) and \(\Delta p\) is larger (from relative velocity analysis), then \(F_{avg}\) is larger.

Step3: Analyze crumple zone

Crumple zones collapse in both front (head - on) and back (rear - end) collisions. So the statement “The collapse of the crumple zone in the front of the car occurs only in the head - on collision” is wrong.

Step4: Analyze momentum and velocity relative to ground

If we assume the initial speed of the car (and driver) before collision (in a multi - car scenario) is same (e.g., on a highway, all cars are moving at some speed \(v_{0}\) before the accident). In a head - on, another car is coming at \(v_{0}\) (opposite direction), but the driver's initial momentum relative to ground is \(p = mv_{0}\) (same as before rear - end if initial \(v_{0}\) is same). Velocity relative to ground initially is \(v_{0}\) (same for both cases before collision).

Answer:

The change in momentum is greater in the head - on collision. The average force on the driver is greater in the head - on collision.