Question

1. engineers perform a crash test with a minivan and a compact car. the mass of the minivan is ( 8.0\times10^{2}\text{ kg} ), and the mass of the compact car is ( 560\text{ kg} ). the minivan was moving west, and the compact car was moving south. after the collision, the two cars crumpled together and moved at ( 15\text{ m/s w }30^{circ}\text{ s} ). calculate the initial velocity of each vehicle.

Explanation:

Step1: Calculate the total mass after collision

The total mass \(m_{total}=m_{minivan}+m_{compact}\)
\(m_{minivan} = 800\space kg\), \(m_{compact}=560\space kg\)
\(m_{total}=800 + 560=1360\space kg\)

Step2: Analyze the momentum in the west - direction

Let the initial velocity of the minivan be \(v_{minivan}\) (in the west - direction).
The momentum in the west - direction before collision \(p_{west - initial}=m_{minivan}v_{minivan}\)
The momentum in the west - direction after collision \(p_{west - final}=m_{total}v_{final}\cos30^{\circ}\)
By conservation of momentum \(m_{minivan}v_{minivan}=m_{total}v_{final}\cos30^{\circ}\)
Substitute \(m_{minivan} = 800\space kg\), \(m_{total}=1360\space kg\), \(v_{final}=15\space m/s\)
\(v_{minivan}=\frac{m_{total}v_{final}\cos30^{\circ}}{m_{minivan}}\)
\(v_{minivan}=\frac{1360\times15\times\frac{\sqrt{3}}{2}}{800}\)
\(v_{minivan}=\frac{1360\times15\times0.866}{800}\)
\(v_{minivan}=\frac{1360\times12.99}{800}\)
\(v_{minivan}=\frac{17666.4}{800}=22.083\approx22\space m/s\) (west)

Step3: Analyze the momentum in the south - direction

Let the initial velocity of the compact car be \(v_{compact}\) (in the south - direction)
The momentum in the south - direction before collision \(p_{south - initial}=m_{compact}v_{compact}\)
The momentum in the south - direction after collision \(p_{south - final}=m_{total}v_{final}\sin30^{\circ}\)
By conservation of momentum \(m_{compact}v_{compact}=m_{total}v_{final}\sin30^{\circ}\)
Substitute \(m_{compact}=560\space kg\), \(m_{total}=1360\space kg\), \(v_{final}=15\space m/s\)
\(v_{compact}=\frac{m_{total}v_{final}\sin30^{\circ}}{m_{compact}}\)
\(v_{compact}=\frac{1360\times15\times0.5}{560}\)
\(v_{compact}=\frac{1360\times7.5}{560}\)
\(v_{compact}=\frac{10200}{560}\approx18.21\approx18\space m/s\) (south)

Answer:

The initial velocity of the minivan is approximately \(22\space m/s\) (west) and the initial velocity of the compact car is approximately \(18\space m/s\) (south)