A ball is allowed to fall from a height of 10m . If there is 40% loss of energy due to impact, then after one impact ball will go up to

Options

A. 10 m
B. 8 m
C. 4 m
D. 6 m

Answer

To solve the problem step by step, we will analyze the situation of the ball falling from a height and the energy transformations that occur during the fall and after the impact.

### Step 1: Calculate the potential energy at the height of 10 meters.
The potential energy (PE) of the ball at a height \( h \) is given by the formula:
\[ PE = mgh \]
Where:
- \( m \) = mass of the ball (we will see that it cancels out later)
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- \( h \) = height (10 m)

Substituting the values:
\[ PE = mg \cdot 10 \]

### Step 2: Determine the kinetic energy just before impact.
As the ball falls, the potential energy is converted into kinetic energy (KE) just before it hits the ground. Therefore, the kinetic energy just before impact is:
\[ KE = PE = mg \cdot 10 \]

### Step 3: Calculate the kinetic energy after the impact.
According to the problem, there is a 40% loss of energy due to the impact. This means that the ball retains 60% of its kinetic energy after the impact. Thus, the kinetic energy after the impact \( KE' \) is:
\[ KE' = 60\% \, \text{of} \, KE = 0.6 \cdot KE = 0.6 \cdot (mg \cdot 10) = 6mg \]

### Step 4: Relate the kinetic energy after impact to the height it reaches.
After the impact, the kinetic energy \( KE' \) will be converted back into potential energy as the ball rises. The potential energy at the new height \( h \) is given by:
\[ PE' = mgh \]
Setting this equal to the kinetic energy after the impact:
\[ mgh = 6mg \]

### Step 5: Cancel out the mass and solve for height \( h \).
Since \( m \) appears on both sides of the equation, we can cancel it out:
\[ gh = 6g \]
Now, dividing both sides by \( g \):
\[ h = 6 \, \text{meters} \]

### Conclusion
The ball will rise to a height of 6 meters after the impact.

### Final Answer
The ball will go up to **6 meters** after one impact.

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