Questions: In an automobile crash, a vehicle that was stopped at a red light is rear-ended by another vehicle. The vehicles have the same mass. If the tire marks show that the two vehicles moved after the collision at 4 m / s, what was the speed of the vehicle before the collision?

In an automobile crash, a vehicle that was stopped at a red light is rear-ended by another vehicle. The vehicles have the same mass. If the tire marks show that the two vehicles moved after the collision at 4 m / s, what was the speed of the vehicle before the collision?
Transcript text: 3. In an automobile crash, a vehicle that was stopped at a red light is rear-ended by another vehicle. The vehicles have the same mass. If the tire marks show that the two vehicles moved after the collision at $4 \mathrm{~m} / \mathrm{s}$, what was the speed of the vehicle before the collision?
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Solution

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Solution Steps

Step 1: Understand the Problem

We are dealing with a collision problem where two vehicles of equal mass collide, and we need to find the speed of the moving vehicle before the collision. The vehicles move together at \(4 \, \text{m/s}\) after the collision.

Step 2: Apply Conservation of Momentum

Since the collision is between two vehicles of equal mass and they move together after the collision, we can use the conservation of momentum. The total momentum before the collision equals the total momentum after the collision.

Step 3: Set Up the Equation

Let \( m \) be the mass of each vehicle, \( v \) be the speed of the moving vehicle before the collision, and \( v_f = 4 \, \text{m/s} \) be the speed of both vehicles after the collision. The vehicle at rest has an initial speed of \(0 \, \text{m/s}\).

The equation for conservation of momentum is: \[ m \cdot v + m \cdot 0 = (m + m) \cdot v_f \]

Step 4: Solve for the Initial Speed

Simplify the equation: \[ m \cdot v = 2m \cdot v_f \]

Divide both sides by \( m \): \[ v = 2 \cdot v_f \]

Substitute \( v_f = 4 \, \text{m/s} \): \[ v = 2 \cdot 4 \, \text{m/s} = 8 \, \text{m/s} \]

Final Answer

\(\boxed{8 \, \text{m/s}}\)

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