In a perfectly inelastic collision, the two objects stick together and move as one after the collision. The energy loss in such a collision is calculated as the difference between the initial kinetic energy and the final kinetic energy.

Step 1: Calculate the initial kinetic energy
The initial kinetic energy (KE_initial) is the sum of the kinetic energy of both objects before the collision. Kinetic energy is given by the formula KE = 1/2 * m * v^2, where m is the mass and v is the velocity.

For the first object:
KE1_initial = 1/2 * 40kg * (4m/s)^2 = 320 Joules

For the second object:
KE2_initial = 1/2 * 60kg * (2m/s)^2 = 120 Joules

So, KE_initial = KE1_initial + KE2_initial = 320J + 120J = 440 Joules

Step 2: Calculate the final kinetic energy
After the collision, the two objects move together with a common velocity. This can be found using the conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision. Momentum is given by the formula p = m * v.

The total initial momentum (p_initial) is the sum of the momentum of both objects before the collision:

p_initial = 40kg * 4m/s + 60kg * 2m/s = 280 kg*m/s

Since the total momentum is conserved, the total final momentum (p_final) is also 280 kg*m/s. As the two objects move together after the collision, their total mass is 40kg + 60kg = 100kg. Therefore, their common velocity (v_common) can be found as:

v_common = p_final / (m1 + m2) = 280 kg*m/s / 100kg = 2.8 m/s

The final kinetic energy (KE_final) is then:

KE_final = 1/2 * (m1 + m2) * v_common^2 = 1/2 * 100kg * (2.8m/s)^2 = 392 Joules

Step 3: Calculate the energy loss
The energy loss is the difference between the initial and final kinetic energy:

Energy loss = KE_initial - KE_final = 440J - 392J = 48 Joules

So, the energy loss in this perfectly inelastic collision is 48 Joules.

Question

In a perfectly inelastic collision, the two objects stick together and move as one after the collision. The energy loss in such a collision is calculated as the difference between the initial kinetic energy and the final kinetic energy.

Step 1: Calculate the initial kinetic energy
The initial kinetic energy (KE_initial) is the sum of the kinetic energy of both objects before the collision. Kinetic energy is given by the formula KE = 1/2 * m * v^2, where m is the mass and v is the velocity.

For the first object:
KE1_initial = 1/2 * 40kg * (4m/s)^2 = 320 Joules

For the second object:
KE2_initial = 1/2 * 60kg * (2m/s)^2 = 120 Joules

So, KE_initial = KE1_initial + KE2_initial = 320J + 120J = 440 Joules

Step 2: Calculate the final kinetic energy
After the collision, the two objects move together with a common velocity. This can be found using the conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision. Momentum is given by the formula p = m * v.

The total initial momentum (p_initial) is the sum of the momentum of both objects before the collision:

p_initial = 40kg * 4m/s + 60kg * 2m/s = 280 kg*m/s

Since the total momentum is conserved, the total final momentum (p_final) is also 280 kg*m/s. As the two objects move together after the collision, their total mass is 40kg + 60kg = 100kg. Therefore, their common velocity (v_common) can be found as:

v_common = p_final / (m1 + m2) = 280 kg*m/s / 100kg = 2.8 m/s

The final kinetic energy (KE_final) is then:

KE_final = 1/2 * (m1 + m2) * v_common^2 = 1/2 * 100kg * (2.8m/s)^2 = 392 Joules

Step 3: Calculate the energy loss
The energy loss is the difference between the initial and final kinetic energy:

Energy loss = KE_initial - KE_final = 440J - 392J = 48 Joules

So, the energy loss in this perfectly inelastic collision is 48 Joules.

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