A ball whose mass is 100g is dropped from a height of 2m from the floor. it rebounds vertically upwards after colliding with the floor to a height 1.5 m. Find(a)The momentum of the ball before and after collision,(b)the average force exerted by the floor on the ball.Assume that collision lasts for 10^-8secs.
Solution
To solve this problem, we can use the principles of conservation of momentum and the impulse-momentum theorem.
(a) Finding the momentum of the ball before and after collision:
Momentum is defined as the product of an object's mass and velocity. In this case, we are given the mass of the ball (100g), but we need to find its initial and final velocities.
First, let's convert the mass of the ball to kilograms: 100g = 0.1kg.
To find the initial velocity of the ball, we can use the equation of motion for vertical free fall:
h = (1/2) * g * t^2
Where h is the height, g is the acceleration due to gravity, and t is the time.
Rearranging the equation, we can solve for time:
t = sqrt((2h) / g)
t = sqrt((2 * 2m) / 9.8m/s^2) ≈ 0.640s
Using the equation of motion for free fall:
v = g * t
v = 9.8m/s^2 * 0.640s ≈ 6.272m/s
Now we can calculate the momentum before collision:
Momentum before collision = mass * velocity
P1 = 0.1kg * 6.272m/s = 0.6272 kg·m/s
Next, let's find the final velocity of the ball after collision. Since it rebounds vertically upwards, the final velocity will be negative (opposite direction).
Using the equation of motion:
v = u + a * t
Where v is the final velocity, u is the initial velocity, a is the acceleration due to gravity (-9.8m/s^2), and t is the time (0.640s).
v = u + a * t
v = 0 + (-9.8m/s^2) * 0.640s
v ≈ -6.272m/s
Now we can calculate the momentum after collision:
Momentum after collision = mass * velocity
P2 = 0.1kg * (-6.272m/s) = -0.6272 kg·m/s
(b) Finding the average force exerted by the floor on the ball:
The impulse-momentum theorem states that the change in momentum of an object is equal to the impulse applied to it.
Impulse = Force * Time
The average force exerted by the floor on the ball can be determined by calculating the change in momentum and dividing it by the time of collision.
Change in momentum = P2 - P1
ΔP = (-0.6272 kg·m/s) - (0.6272 kg·m/s) = -1.2544 kg·m/s
Time of collision = 10^-8 s
Average force exerted by the floor = ΔP / Time of collision
Force = -1.2544 kg·m/s / 10^-8 s
Force = -1.2544 * (1/10^-8) N
Force ≈ -1.2544 * 10^8 N
Therefore, the average force exerted by the floor on the ball is approximately -1.2544 * 10^8 Newtons.