A 1.5 kg
basketball is thrown to the floor. At the moment it first hits the floor, the basketball has a
velocity of 6 m/s directed 45° below the horizontal. In contact with the ball for 0.1 seconds, the
floor exerts upon the ball a force of 135 N directed 110° above the horizontal. What is the
velocity (magnitude and direction) of the ball at the moment it leaves the floor?
Once the ball leaves the floor, it is in projectile motion. How high will the ball bounce, and
through what horizontal distance will the ball move, before hitting the floor again?
To solve this problem, we need to break it down into different parts and analyze each part separately.
1. Calculating the velocity of the ball as it leaves the floor:
- The initial velocity of the ball before it hits the floor is given as 6 m/s at an angle of 45° below the horizontal.
- The force exerted by the floor on the ball is given as 135 N at an angle of 110° above the horizontal.
- The time of contact between the ball and the floor is 0.1 seconds.
To find the final velocity of the ball as it leaves the floor, we need to use the concept of impulse.
Impulse is defined as the change in momentum of an object. In this case, the impulse is equal to the force applied multiplied by the time of contact:
Impulse = Force × Time
The change in momentum is given by:
Change in momentum = Mass × (Final velocity - Initial velocity)
Since the mass of the basketball is given as 1.5 kg, we can use the above formulas to solve for the final velocity.
2. Analyzing the projectile motion after leaving the floor:
- Once the ball leaves the floor, it is in projectile motion. This means that it will follow a parabolic path.
- The maximum height reached by the ball can be determined using the kinematic equation for vertical motion:
Maximum height = (Initial vertical velocity)^2 / (2 × Gravitational acceleration)
- The horizontal distance covered by the ball before hitting the floor again can be determined using the kinematic equation for horizontal motion:
Horizontal distance = Horizontal velocity × Time of flight
The time of flight can be found by dividing the total time taken for the ball to reach the floor again into two halves:
Time of flight = (Total time taken for ball to reach the floor again) / 2
Now, we can use these formulas to calculate the values required.