Before a collision, a 24 kg object is moving at +13 m/s. Find the impulse that acted on the object if, after the collision, it moved at the following velocities.
(a) +8.0 m/s
(b) -8.0 m/s
Impulse = integral F dt = change in momentum = final momentum - initial momentum
24 (8) - 24 (13)
24 (-8) - 24 (13)
To find the impulse that acted on the object, we can use the impulse-momentum principle, which states that the impulse is equal to the change in momentum of the object.
The momentum of an object is calculated by multiplying its mass by its velocity.
Let's proceed step-by-step:
Step 1: Calculate the initial momentum of the object.
Momentum = mass × velocity
Momentum = 24 kg × 13 m/s
Momentum = 312 kg·m/s
(a) If the object moves at +8.0 m/s after the collision:
Step 2: Calculate the final momentum of the object.
Final Momentum = mass × velocity
Final Momentum = 24 kg × 8.0 m/s
Final Momentum = 192 kg·m/s
Step 3: Calculate the change in momentum.
Change in Momentum = Final Momentum - Initial Momentum
Change in Momentum = 192 kg·m/s - 312 kg·m/s
Change in Momentum = -120 kg·m/s
Therefore, the impulse that acted on the object when it moves at +8.0 m/s after the collision is -120 kg·m/s.
(b) If the object moves at -8.0 m/s after the collision:
Step 2: Calculate the final momentum of the object.
Final Momentum = mass × velocity
Final Momentum = 24 kg × (-8.0 m/s)
Final Momentum = -192 kg·m/s
Step 3: Calculate the change in momentum.
Change in Momentum = Final Momentum - Initial Momentum
Change in Momentum = -192 kg·m/s - 312 kg·m/s
Change in Momentum = -504 kg·m/s
Therefore, the impulse that acted on the object when it moves at -8.0 m/s after the collision is -504 kg·m/s.
To find the impulse that acted on the object, we can use the impulse-momentum principle, which states that the change in momentum of an object is equal to the impulse acting on it.
The momentum of an object is given by the product of its mass and velocity: p = m * v.
In this case, the initial momentum of the object is calculated as follows:
p_initial = m * v_initial,
where m is the mass of the object (24 kg), and v_initial is the initial velocity (+13 m/s).
(a) To find the impulse when the final velocity is +8.0 m/s, we need to calculate the final momentum first. The change in momentum is then given by the difference between the final and initial momenta:
p_difference = p_final - p_initial.
The final momentum (p_final) can be calculated as follows:
p_final = m * v_final,
where v_final is the final velocity (+8.0 m/s).
Plugging in the values, we get:
p_initial = 24 kg * 13 m/s = 312 kg·m/s,
p_final = 24 kg * 8.0 m/s = 192 kg·m/s.
Therefore, the impulse is given by the change in momentum:
Impulse = p_difference = p_final - p_initial = 192 kg·m/s - 312 kg·m/s = -120 kg·m/s.
So, the impulse that acted on the object when the final velocity is +8.0 m/s is -120 kg·m/s.
(b) To find the impulse when the final velocity is -8.0 m/s, we follow the same steps as in part (a).
The final momentum is given by:
p_final = m * v_final = 24 kg * -8.0 m/s = -192 kg·m/s.
Calculating the change in momentum:
p_difference = p_final - p_initial = -192 kg·m/s - 312 kg·m/s = -504 kg·m/s.
Hence, the impulse that acted on the object when the final velocity is -8.0 m/s is -504 kg·m/s.