Two cars of equal mass are traveling as shown in the figure below just before undergoing a collision. Before the collision, one of the cars has a speed of 19 m/s along +x, and the other has a speed of 31 m/s along +y. The cars lock bumpers and then slide away together after the collision. What are the magnitude and direction of their final velocity?

Question

Two cars of equal mass are traveling as shown in the figure below just before undergoing a collision. Before the collision, one of the cars has a speed of 19 m/s along +x, and the other has a speed of 31 m/s along +y. The cars lock bumpers and then slide away together after the collision. What are the magnitude and direction of their final velocity?

Answer 1

To find the magnitude and direction of the final velocity after the collision, we can use the principles of momentum conservation. According to the law of conservation of momentum, the total momentum of an isolated system remains constant before and after a collision.

First, let's calculate the initial momentum of the two cars individually. Momentum is defined as the product of mass and velocity.

For the x-direction car:
Mass of car = m1 (assuming equal mass)
Velocity in x-direction = 19 m/s
Momentum in x-direction = m1 * 19

For the y-direction car:
Mass of car = m2 (assuming equal mass)
Velocity in y-direction = 31 m/s
Momentum in y-direction = m2 * 31

Since there is no external force acting on the system, the total initial momentum in the x-direction is equal to the total final momentum in the x-direction. Similarly, the total initial momentum in the y-direction is equal to the total final momentum in the y-direction.

Using the principle of momentum conservation, we can set up two equations:

m1 * 19 = Total final momentum in the x-direction
m2 * 31 = Total final momentum in the y-direction

Now, let's find the total final momentum in the x-direction. Since the cars lock bumpers and slide away together, the final velocity will be in the direction of the resultant vector of their initial velocities.

Using vector addition, we can find the resultant velocity:
Resultant velocity = sqrt((19^2) + (31^2))
Resultant velocity = sqrt(361 + 961)
Resultant velocity = sqrt(1322)
Resultant velocity ≈ 36.36 m/s

The direction of the resultant velocity can be determined using the tangent inverse function:
Direction of resultant velocity = atan(31/19)
Direction of resultant velocity ≈ 59.41° (measured counterclockwise from the positive x-axis)

Therefore, the magnitude of the final velocity after the collision is approximately 36.36 m/s, and the direction is approximately 59.41° counterclockwise from the positive x-axis.