A boy moves a ball of mass 0.5 kg in horizontal surface with 20 m/s. It collides and moves with 5% of its initial kinetic energy. Find final speed

Options

A. `sqrt(5) m/s`
B. `4 sqrt(5) m/s`
C. `2 sqrt(5) m/s`
D. `2 m/s`

Answer

To solve the problem step by step, we will calculate the final speed of the ball after it collides and moves with 5% of its initial kinetic energy.

### Step 1: Calculate the Initial Kinetic Energy (KE_initial)
The formula for kinetic energy is given by:
\[ KE = \frac{1}{2} m v^2 \]
Where:
- \( m = 0.5 \, \text{kg} \) (mass of the ball)
- \( v = 20 \, \text{m/s} \) (initial speed of the ball)

Substituting the values:
\[ KE_{\text{initial}} = \frac{1}{2} \times 0.5 \times (20)^2 \]
\[ KE_{\text{initial}} = \frac{1}{2} \times 0.5 \times 400 \]
\[ KE_{\text{initial}} = 0.25 \times 400 = 100 \, \text{J} \]

### Step 2: Calculate the Final Kinetic Energy (KE_final)
According to the problem, after the collision, the ball moves with 5% of its initial kinetic energy. Therefore:
\[ KE_{\text{final}} = 5\% \times KE_{\text{initial}} = \frac{5}{100} \times 100 \]
\[ KE_{\text{final}} = 5 \, \text{J} \]

### Step 3: Relate Final Kinetic Energy to Final Speed
Using the kinetic energy formula again for the final kinetic energy:
\[ KE_{\text{final}} = \frac{1}{2} m v_f^2 \]
Where \( v_f \) is the final speed we want to find. Setting this equal to the final kinetic energy we calculated:
\[ 5 = \frac{1}{2} \times 0.5 \times v_f^2 \]

### Step 4: Solve for Final Speed \( v_f \)
Rearranging the equation:
\[ 5 = \frac{0.25}{2} v_f^2 \]
\[ 5 = 0.25 v_f^2 \]
\[ v_f^2 = \frac{5}{0.25} = 20 \]
\[ v_f = \sqrt{20} = 2\sqrt{5} \, \text{m/s} \]

### Final Answer
The final speed of the ball after the collision is:
\[ v_f = 2\sqrt{5} \, \text{m/s} \]

---