A rubber ball is dropped from a height of `5m `on a plane, where the acceleration due to gravity is not shown. On bouncing it rises to `1.8 m.` The ball loses its velocity on bouncing by a factor of

Options

A. `16//25`
B. `2//5`
C. `3//5`
D. `9//25`

Answer

To solve the problem of how much velocity the rubber ball loses on bouncing, we can follow these steps:

### Step 1: Determine the initial velocity before impact
When the ball is dropped from a height \( h_1 = 5 \, \text{m} \), we can calculate the velocity just before it hits the ground using the formula for gravitational potential energy converted to kinetic energy:

\[ v_1 = \sqrt{2gh_1} \]

### Step 2: Calculate the height after bouncing
After bouncing, the ball rises to a height \( h_2 = 1.8 \, \text{m} \). The velocity just after the bounce can be calculated using the same formula:

\[ v_2 = \sqrt{2gh_2} \]

### Step 3: Find the ratio of the velocities
To find the factor by which the ball loses its velocity, we need to calculate the ratio of the velocities \( v_2 \) and \( v_1 \):

\[ \frac{v_2}{v_1} = \frac{\sqrt{2gh_2}}{\sqrt{2gh_1}} = \sqrt{\frac{h_2}{h_1}} \]

### Step 4: Substitute the heights into the ratio
Substituting the values of \( h_1 \) and \( h_2 \):

\[ \frac{v_2}{v_1} = \sqrt{\frac{1.8}{5}} = \sqrt{\frac{18}{50}} = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

### Step 5: Calculate the loss in velocity
The loss in velocity can be expressed as:

\[ \text{Loss factor} = 1 - \frac{v_2}{v_1} = 1 - \frac{3}{5} = \frac{2}{5} \]

### Conclusion
The ball loses its velocity on bouncing by a factor of \( \frac{2}{5} \).

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