IMPACT PROBLEM-Please help me understand the answer, I am lost:(

Question

The parking brake on a 2000 kg Cadillac has failed, and it is rolling slowly, at 4 m/s, toward a group of children. Seeing the situation, you realize you have just enough time to drive your 1000 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?

(answer: 8 m/s)

Answer 1

Hi Lauren,

This problem involves the conservation of momentum. Before the collision, the Cadillac has a momentum because it's moving:

\(P_{\text{Cadillac}} = m_{\text{Cadillac}} \cdot v_{\text{Cadillac}}\)

where

\(m_{\text{Cadillac}} = 2000 \, \text{kg}\) (mass of the Cadillac) and

\(v_{\text{Cadillac}} = 4 \, \text{m/s}\) (velocity of the Cadillac).

The Volkswagen is initially at rest, so its momentum is zero:

\(P_{\text{Volkswagen}} = 0\).

After the collision, the combined system of the Cadillac and the Volkswagen should have a total momentum of zero to bring them both to a halt. So:

\(P_{\text{total}} = P_{\text{Cadillac}} + P_{\text{Volkswagen}} = 0\)

Now, you can solve for the velocity (\(v_{\text{Volkswagen}}\)) of the Volkswagen after the collision:

\(P_{\text{Volkswagen}} = -P_{\text{Cadillac}}\)

\(m_{\text{Volkswagen}} \cdot v_{\text{Volkswagen}} = -m_{\text{Cadillac}} \cdot v_{\text{Cadillac}}\)

Substitute the known values:

\(1000 \, \text{kg} \cdot v_{\text{Volkswagen}} = -(2000 \, \text{kg} \cdot 4 \, \text{m/s})\)

Now, solve for \(v_{\text{Volkswagen}}\):

\(v_{\text{Volkswagen}} = \frac{-(2000 \, \text{kg} \cdot 4 \, \text{m/s})}{1000 \, \text{kg}} = -8 \, \text{m/s}\)

Since we're looking for the speed (magnitude), take the absolute value:

\(v_{\text{Volkswagen}} = 8 \, \text{m/s}\).

So, the speed at which you should impact the Cadillac to bring it to a halt is \(8 \, \text{m/s}\).

Hope this help out