A ball of mass 0.2kg falls from a height of 45m. On striking the ground it rebounds in 0.1s with two-thirds of the velocity with which it struck the ground. Calculate i the momentum change on hitting the ground, ii the forcn on the ball due to the impact.
1.10Ns
2.100N
Momentum is 10 Ns.
To calculate the momentum change on hitting the ground, you need to find the initial momentum (before hitting the ground) and the final momentum (after hitting the ground).
i) Initial Momentum:
The initial momentum of an object is given by the formula:
Initial momentum = mass * initial velocity
Given:
Mass of the ball (m) = 0.2 kg
Initial velocity (u) = ? (not given directly)
To find the initial velocity, we need to use the concept of conservation of mechanical energy.
The potential energy at the initial height (before falling) gets converted into kinetic energy when the ball reaches the ground.
The potential energy (PE) at height (h) is given by the formula:
PE = mass * gravitational acceleration * height
Given:
Height (h) = 45 m
Gravitational acceleration (g) = 9.8 m/s² (approximate value)
Substituting the values, we have:
PE = 0.2 kg * 9.8 m/s² * 45 m
PE = 88.2 J (Joules)
Since the potential energy gets converted into kinetic energy, we can equate these two values:
PE = KE
88.2 J = (1/2) * mass * initial velocity²
Rearranging the equation, we can find the initial velocity:
Initial velocity = √(2 * PE / mass)
Initial velocity = √(2 * 88.2 J / 0.2 kg)
Initial velocity = √(882 J / 0.2 kg)
Initial velocity = √(4410 m²/s² / kg)
Initial velocity ≈ √(4410 m²/s²) ≈ 66.4 m/s (approximate value)
ii) Final Momentum:
The final momentum of an object is given by the formula:
Final momentum = mass * final velocity
Given:
Final velocity (v) = (2/3) * initial velocity (as stated in the problem)
Final velocity = (2/3) * 66.4 m/s
Final velocity ≈ 44.3 m/s (approximate value)
Now we can calculate the momentum change on hitting the ground:
Momentum change = Final momentum - Initial momentum
Momentum change = (mass * final velocity) - (mass * initial velocity)
Momentum change = mass * (final velocity - initial velocity)
Momentum change = 0.2 kg * (44.3 m/s - 66.4 m/s)
Momentum change = 0.2 kg * (-22.1 m/s)
Momentum change ≈ -4.42 kg·m/s (approximate value)
ii) Force on the ball due to the impact:
Force can be calculated using the formula:
Force = rate of change of momentum
Given:
Time taken for the rebound (t) = 0.1 s
Using the formula, we can calculate the force:
Force = Momentum change / time
Force = (-4.42 kg·m/s) / (0.1 s)
Force ≈ -44.2 N (approximate value)
Note: The negative sign indicates that the direction of force is opposite to the direction of motion.
Therefore, the answers to your questions are:
i) The momentum change on hitting the ground is approximately -4.42 kg·m/s.
ii) The force on the ball due to the impact is approximately -44.2 N.