To make a bounce pass, a player throws a 0.60-kg basketball toward the floor. The ball hits the floor with a speed of 6.5 m/s at an angle of 57° from the vertical. If the ball rebounds with the same speed and angle, what was the impulse delivered to it by the floor?
To find the impulse delivered to the basketball by the floor, we will use the impulse-momentum theorem.
The impulse is given by the change in momentum of the basketball. In this case, the momentum before and after the bounce is the same since the ball rebounds with the same speed and angle. Therefore, the impulse is equal to zero.
Impulse (𝐽) = Change in momentum (𝑝)
Since the impulse is zero, this implies that the change in momentum is also zero.
Change in momentum (𝑝) = 0
Therefore, the impulse delivered to the basketball by the floor is zero.
To find the impulse delivered to the ball by the floor, we need to calculate the change in momentum of the ball. Impulse is defined as the change in momentum of an object.
The momentum of an object is given by the product of its mass and velocity. Therefore, the initial momentum of the ball can be calculated as the product of its mass (0.60 kg) and initial velocity (6.5 m/s).
The initial momentum of the ball is:
Initial momentum = mass * initial velocity
= 0.60 kg * 6.5 m/s
= 3.9 kg·m/s
Now, let's calculate the final momentum of the ball. Since the ball rebounds with the same speed and angle, the final vertical velocity will be equal in magnitude but opposite in direction. Therefore, the final horizontal velocity will remain the same.
The final momentum of the ball is:
Final momentum = mass * final velocity
To calculate the final velocity, we need to separate the motion into horizontal and vertical components. Since the vertical velocity after the bounce is the same in magnitude but opposite in direction, we can determine the vertical component using trigonometry.
Vertical component of velocity = initial velocity * sin(angle)
= 6.5 m/s * sin(57°)
≈ 5.525 m/s
The final vertical velocity after the bounce will be -5.525 m/s.
Now, for the horizontal component, the velocity remains the same:
Horizontal component of velocity = initial velocity * cos(angle)
= 6.5 m/s * cos(57°)
≈ 3.043 m/s
The final velocity after the bounce will be 3.043 m/s horizontally and -5.525 m/s vertically.
Now, let's calculate the final momentum:
Final momentum = mass * final velocity
= 0.60 kg * (3.043 m/s + (-5.525 m/s)) (considering the directions)
= 0.60 kg * (-2.482 m/s)
≈ -1.489 kg·m/s
To calculate the impulse, we need to find the change in momentum, which is simply the difference between the final momentum and initial momentum:
Impulse = Final momentum - Initial momentum
= -1.489 kg·m/s - 3.9 kg·m/s
≈ -5.389 kg·m/s
Therefore, the impulse delivered to the ball by the floor is approximately -5.389 kg·m/s. The negative sign indicates that the impulse is in the opposite direction to the initial momentum of the ball.