A 1920-kg Cadillac and a 1000-kg Volkswagen meet at an intersection. The stoplight has just turned green, and the Cadillac, heading north, drives forward into the intersection. The Volkswagen, traveling east, fails to stop. The Volkswagen crashes into the left front fender of the Cadillac; then the cars stick together and slide to a halt. Officer Tom, responding to the accident, sees that the skid marks are directed 59° north of east from the point of impact. The driver of the Cadillac, who keeps a close eye on the speedometer, reports that he was traveling at 31 m/s when the accident occurred. How fast was the Volkswagen going just before the impact?
To solve this problem, we can use the principles of conservation of momentum and vector analysis.
1. First, let's calculate the momentum of the Cadillac just before the collision using the formula: momentum = mass x velocity.
Momentum of the Cadillac = Mass of the Cadillac * Velocity of the Cadillac = 1920 kg * 31 m/s = 59520 kg·m/s.
2. Next, we'll need to determine the direction of the momentum vector. Since the Cadillac was heading north, the momentum vector is directed north.
3. Now, let's analyze the collision between the Cadillac and the Volkswagen. The two cars stick together and slide to a halt, so we can assume that momentum is conserved in the collision.
4. From the given information, we know that the skid marks left by the collision are directed 59° north of east. To find the velocity of the Volkswagen just before the impact, we need to determine the component of the Volkswagen's velocity that is directed east.
5. Using trigonometry, we can find the eastward component of the Volkswagen's velocity. Since the angle between the skid marks and the east direction is 59°, the eastward component can be found using the formula: eastward component = velocity * cos(angle).
Eastward component = Velocity of the Volkswagen * cos(59°).
6. Since momentum is conserved, the total momentum after the collision is equal to the momentum of the Cadillac just before the collision. This allows us to set up an equation:
Momentum before collision = Momentum after collision.
Mass of the Cadillac * Velocity of the Cadillac = (Mass of the Cadillac + Mass of the Volkswagen) * Velocity after the collision.
7. Since the two cars stick together after the collision, their combined mass is the sum of the individual masses: (Mass of the Cadillac + Mass of the Volkswagen) = 1920 kg + 1000 kg = 2920 kg.
8. Using the above information, let's set up the equation and solve for the velocity of the Volkswagen just before the impact:
1920 kg * 31 m/s = 2920 kg * Velocity after the collision.
9. Solve the equation to find the velocity after the collision:
Velocity after the collision = (1920 kg * 31 m/s) / 2920 kg.
10. Finally, substitute the value of Velocity after the collision into the equation from step 5 to find the eastward component of the Volkswagen's velocity just before the impact:
Eastward component = Velocity after the collision * cos(59°).
By following these steps, you can determine the velocity of the Volkswagen just before the impact.