After falling from rest at a height of 31.5 m, a 0.564 kg ball rebounds upward, reaching a height of 20.2 m. If the contact between ball and ground lasted 1.80 ms, what average force was exerted on the ball?
To find the average force exerted on the ball, we can use the concept of impulse. The formula for impulse is given by:
Impulse = Force × Time
Impulse is defined as the change in momentum of an object. In this case, the ball falls from rest, rebounds upward, and changes its momentum.
First, let's calculate the initial velocity of the ball before it hits the ground. We can use the equation of motion for free fall:
h = (1/2)gt²
Where h is the height, g is the acceleration due to gravity, and t is the time taken.
Rearranging the equation, we get:
t = √(2h / g)
Plugging in the values, we find:
t = √(2(31.5 m) / 9.8 m/s²)
t ≈ √(64.29)
t ≈ 8.02 s
So, the time taken for the ball to fall is approximately 8.02 seconds.
Now, let's calculate the final velocity of the ball after it rebounds. The final velocity can be calculated using the equation of motion for uniformly accelerated motion:
v = u + at
Where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken.
In this case, the ball is moving upward, so the acceleration will be the acceleration due to gravity, but in the opposite direction (negative). Also, the initial velocity is zero since the ball is at rest.
Using these values, the equation becomes:
20.2 m/s = 0 + (-9.8 m/s²) * t
Simplifying, we find:
20.2 m/s = -9.8 m/s² * t
t = 20.2 m/s / (-9.8 m/s²)
t ≈ -2.06 s
So, the time taken for the ball to reach a height of 20.2 meters is approximately -2.06 seconds.
Since we are given that the contact time between the ball and the ground is 1.80 ms, we can convert it to seconds:
Contact time = 1.80 ms = 1.80 x 10^-3 s
Now, let's calculate the change in momentum of the ball using the equation:
Change in momentum = (final momentum) - (initial momentum)
Momentum is given by the formula:
Momentum = mass × velocity
The initial momentum is mass × initial velocity, and the final momentum is mass × final velocity.
Plugging in the values, we find:
Initial momentum = 0.564 kg × 0 = 0 kg m/s (since the initial velocity is zero)
Final momentum = 0.564 kg × (-9.8 m/s²) = -5.47 kg m/s (since the final velocity is negative)
The change in momentum is:
Change in momentum = (-5.47 kg m/s) - (0 kg m/s)
Change in momentum = -5.47 kg m/s
Now, we can use the definition of impulse to find the average force:
Impulse = Force × Time
The impulse is equal to the change in momentum, so we have:
Force × Time = Change in momentum
Plugging in the values, we find:
Force × (1.80 x 10^-3 s) = -5.47 kg m/s
Simplifying, we find:
Force = (-5.47 kg m/s) / (1.80 x 10^-3 s)
Evaluating the expression, we get:
Force ≈ -3,039.44 N
The negative sign indicates that the force is exerted in the opposite direction (upward) to that of the ball's motion.
Therefore, the average force exerted on the ball is approximately 3,039.44 N, directed upward.