The parking brake on a 1,000 kg Cadillac has failed, and it is rolling slowly at 1 mph toward a...

Question:

The parking brake on a 1,000 kg Cadillac has failed, and it is rolling slowly at 1 mph toward a group of small children. Seeing the situation, you realize you have just enought time to drive your 1,000 kg Volkswagen head-on into the Cadillac and save the children. With what speed should you impact the Cadillac to bring it to a halt?

Conservation of Momentum:

Problems dealing with collisions between objects often require use of conservation of momentum. Momentum is defined as the mass of an object times its velocity: {eq}p = mv {/eq}. It is a law of nature that, in a closed system, the total momentum is conserved: if you add up {eq}p = mv {/eq} for every object, you should get the same answer at every time.

We want the final velocity of the Cadillac to be zero: {eq}v_{Cf} = 0 {/eq}. If we are driving toward the Cadillac head-on, then our final velocity must also be zero: {eq}v_{Vf} = 0 {/eq}. (If the final velocity were positive, then the Volkswagen would begin to roll backward toward the children, defeating the purpose of driving it in the first place; if the final velocity were negative, the Volkswagen would continue to move through the Cadillac, like a ghost.)

Since neither car is moving at the end of this problem, we can say that the final momentum is zero: {eq}p_f = 0 {/eq}. But, by conservation of momentum, this also means that the initial momentum is zero: {eq}p_i = p_f = 0 {/eq}.

The initial momentum is the sum of the Cadillac's momentum and the Volkswagen's momentum. We have both cars' masses and the initial velocity of the Cadillac; this gives us enough information to solve for the Volkswagen's initial volume:

{eq}p_i = 0 = m_C v_C + m_V + v_V\\ 0 = (1000)(1) + (1000)(v_V)\\ -1000 = 1000v_V\\ -1 mph = v_V {/eq}

(Our answer is negative because the Volkswagen is moving in the opposite direction of the Cadillac.)

So, we should drive the car at {eq}1 mph {/eq} to bring the Cadillac to a halt.