You drop a 0.22-kg ball to the floor from a height of 2.3 m , and it bounces to a height of 1.1 m .
What is the magnitude of the change in its momentum as a result of the bounce?
To find the magnitude of the change in momentum, we need to first calculate the initial and final velocities of the ball.
To find the initial velocity, we can use the formula for free fall:
v_initial = sqrt(2 * g * h)
Where:
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = initial height (2.3 m)
v_initial = sqrt(2 * 9.8 * 2.3)
v_initial = 6.95 m/s
Next, we can use the principle of conservation of energy to find the final velocity of the ball. The principle states that the total mechanical energy of an object - the sum of its kinetic energy and potential energy - remains constant, assuming there are no external forces or losses due to friction.
At the initial height, the total mechanical energy is given by:
E_initial = m * g * h
Where:
m = mass of the ball (0.22 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = initial height (2.3 m)
E_initial = 0.22 * 9.8 * 2.3
At the final height, the total mechanical energy is given by:
E_final = m * g * h
Where:
m = mass of the ball (0.22 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = final height (1.1 m)
E_final = 0.22 * 9.8 * 1.1
Since the total mechanical energy remains constant, we have:
E_initial = E_final
Therefore, we can equate the expressions for the initial and final energies:
0.22 * 9.8 * 2.3 = 0.22 * 9.8 * 1.1
Simplifying this equation, we find:
2.3 = 1.1
This result indicates that there seems to be an error in the problem statement because the initial and final heights cannot be equal.
However, assuming the problem statement is correct and there is a typo, we can calculate the final velocity assuming the ball reaches a height of 1.1 m.
v_final = sqrt(2 * g * h)
Where:
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = final height (1.1 m)
v_final = sqrt(2 * 9.8 * 1.1)
v_final = 4.15 m/s
Now that we have the initial velocity (6.95 m/s) and the final velocity (4.15 m/s), we can calculate the change in momentum using the formula:
Change in momentum = mass * (final velocity - initial velocity)
Change in momentum = 0.22 * (4.15 - 6.95)
Change in momentum = -0.57 kg·m/s
The magnitude of the change in momentum is 0.57 kg·m/s.