The stoplight had just changed and a 2000 kg Cadillac had entered the intersection, heading north at 3.0 m/s , when it was struck by a 1000 kg eastbound Volkswagen. The cars stuck together and slid to a halt, leaving skid marks angled 35° north of east. How fast was the Volkswagen going just before the impact?

Verified step by step guidance

1. Step 1: Identify the type of collision. This is an inelastic collision because the two cars stick together after the impact. Momentum is conserved in both the north-south and east-west directions.
2. Step 2: Write the conservation of momentum equations for both directions. For the north-south direction: \( m_{Cadillac} v_{Cadillac} = (m_{Cadillac} + m_{Volkswagen}) v_{final, north} \). For the east-west direction: \( m_{Volkswagen} v_{Volkswagen} = (m_{Cadillac} + m_{Volkswagen}) v_{final, east} \).
3. Step 3: Use the angle of the skid marks (35° north of east) to relate the final velocities in the north and east directions. The relationship is \( \tan(\theta) = \frac{v_{final, north}}{v_{final, east}} \), where \( \theta = 35° \).
4. Step 4: Solve for the final velocities \( v_{final, north} \) and \( v_{final, east} \) using the conservation of momentum equations and the angle relationship. Substitute the known values: \( m_{Cadillac} = 2000 \ \text{kg}, \ m_{Volkswagen} = 1000 \ \text{kg}, \ v_{Cadillac} = 3.0 \ \text{m/s} \).
5. Step 5: Rearrange the equations to solve for the initial velocity of the Volkswagen \( v_{Volkswagen} \). Use the calculated final velocities and the conservation of momentum equations to find \( v_{Volkswagen} \).

Key Concepts

Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum before an event must equal the total momentum after the event, provided no external forces act on it. In collisions, this principle allows us to analyze the velocities of colliding objects by equating their combined momentum before and after the impact.

Vector Components

In physics, vectors have both magnitude and direction, which can be broken down into components along the axes of a coordinate system. For this problem, the velocities of the Cadillac and Volkswagen must be resolved into their respective northward and eastward components to accurately apply the conservation of momentum in two dimensions.

Collision Types

Collisions can be classified as elastic or inelastic. Inelastic collisions, like the one described, involve objects sticking together after impact, resulting in a loss of kinetic energy. Understanding this distinction is crucial for applying the conservation of momentum correctly, as it affects how we calculate the final velocities of the combined mass.