two spheres of mass 3kg and 2kg collide head-on.Their relative velocities before and after collision is 15m/s and 5m/s respectively.Find the loss of kinetic energy.

To find the loss of kinetic energy during the collision of the two spheres, we first need to calculate their initial and final kinetic energies. The kinetic energy (KE) of an object is given by the formula:

Kinetic Energy Formula

The formula for kinetic energy is:

KE = 0.5 * m * v²

where m is the mass of the object and v is its velocity.

Initial Kinetic Energy Calculation

Let's calculate the initial kinetic energies of both spheres before the collision:

• Mass of Sphere 1 (m1): 3 kg
• Velocity of Sphere 1 (v1): 15 m/s
• Mass of Sphere 2 (m2): 2 kg
• Velocity of Sphere 2 (v2): -15 m/s (since it's head-on, we take it as negative)

Now, we can calculate the kinetic energy for each sphere:

For Sphere 1:

KE1 = 0.5 * m1 * v1²

KE1 = 0.5 * 3 kg * (15 m/s)² = 0.5 * 3 * 225 = 337.5 J

For Sphere 2:

KE2 = 0.5 * m2 * v2²

KE2 = 0.5 * 2 kg * (-15 m/s)² = 0.5 * 2 * 225 = 225 J

Total Initial Kinetic Energy

The total initial kinetic energy (KE_initial) is the sum of the kinetic energies of both spheres:

KE_initial = KE1 + KE2 = 337.5 J + 225 J = 562.5 J

Final Kinetic Energy Calculation

Next, we calculate the final kinetic energies after the collision. The relative velocity after the collision is given as 5 m/s. We need to determine the final velocities of both spheres. For simplicity, let’s assume they move in the same direction after the collision:

• Final Velocity of Sphere 1 (v1'): 5 m/s
• Final Velocity of Sphere 2 (v2'): 0 m/s (assuming it comes to rest after the collision)

For Sphere 1:

KE1' = 0.5 * m1 * (v1')²

KE1' = 0.5 * 3 kg * (5 m/s)² = 0.5 * 3 * 25 = 37.5 J

For Sphere 2:

KE2' = 0.5 * m2 * (v2')²

KE2' = 0.5 * 2 kg * (0 m/s)² = 0 J

Total Final Kinetic Energy

The total final kinetic energy (KE_final) is the sum of the kinetic energies of both spheres after the collision:

KE_final = KE1' + KE2' = 37.5 J + 0 J = 37.5 J

Calculating the Loss of Kinetic Energy

The loss of kinetic energy during the collision can be calculated by subtracting the final kinetic energy from the initial kinetic energy:

Loss of KE = KE_initial - KE_final

Loss of KE = 562.5 J - 37.5 J = 525 J

In summary, the loss of kinetic energy during the collision of the two spheres is 525 Joules. This significant loss indicates that a considerable amount of energy was transformed into other forms, likely due to factors such as deformation or heat during the collision.