A rubber ball is dropped from a height of 5m on a plane. On bouncing it rises to 1.8m. The ball loses its velocity on bouncing by a factor of –

• A) &dfrac{16}{25}
• B) &dfrac{2}{5}
• C) &dfrac{3}{5}
• D) &dfrac{9}{25}

To determine how much velocity the rubber ball loses upon bouncing, we need to analyze the situation using the principles of energy conservation and the relationship between height and velocity. When the ball is dropped from a height, it converts potential energy into kinetic energy as it falls. Upon bouncing, it doesn't return to its original height, indicating that some energy is lost, likely due to factors like air resistance and inelastic deformation.

Understanding the Relationship Between Height and Velocity

When an object is dropped from a height, the potential energy at that height is given by the formula:

Potential Energy (PE) = mgh

where m is mass, g is the acceleration due to gravity (approximately 9.81 m/s²), and h is the height. As the ball falls, this potential energy converts into kinetic energy (KE) just before it hits the ground:

Kinetic Energy (KE) = ½ mv²

At the maximum height after bouncing, the kinetic energy converts back into potential energy. The height the ball reaches after bouncing is 1.8 m. We can set up the following equations to find the velocities:

Calculating Initial and Final Velocities

1. **Initial Drop from 5 m**:

Using the potential energy at 5 m:

PE_initial = mgh_initial = mg(5)

Just before hitting the ground, all this potential energy converts to kinetic energy:

KE_initial = ½ mv_initial² = mg(5)

Thus, we can equate and solve for the initial velocity (v_initial):

mg(5) = ½ mv_initial²

Canceling mass (m) from both sides:

g(5) = ½ v_initial²

v_initial² = 10g

v_initial = √(10g) = √(10 × 9.81) ≈ 9.9 m/s

2. **After Bouncing to 1.8 m**:

Using the same approach for the height of 1.8 m:

PE_final = mgh_final = mg(1.8)

At the peak after bouncing, this potential energy equals the kinetic energy just after the bounce:

KE_final = ½ mv_final² = mg(1.8)

Equating gives:

mg(1.8) = ½ mv_final²

Canceling mass (m):

g(1.8) = ½ v_final²

v_final² = 3.6g

v_final = √(3.6g) = √(3.6 × 9.81) ≈ 6.0 m/s

Calculating the Loss of Velocity

Now that we have both velocities, we can find the factor by which the velocity decreases:

Loss of velocity factor = v_final / v_initial = 6.0 / 9.9

Calculating this gives:

Loss of velocity factor ≈ 0.606 (approximately)

Finding the Velocity Loss Factor

To express the loss in velocity as a fraction, we can calculate:

Loss in velocity = v_initial - v_final = 9.9 - 6.0 = 3.9 m/s

Now, the factor of velocity lost can be calculated as:

Loss factor = (v_initial - v_final) / v_initial = 3.9 / 9.9 ≈ 0.3939

To express this as a fraction, we can simplify it:

0.3939 is approximately equal to &frac{3}{5}.

Final Answer

Therefore, the ball loses its velocity on bouncing by a factor of &frac{3}{5}, which corresponds to option C.