An inelastic ball is dropped from a height 100 meter. If due to impact it loses 35% of its energy the ball will rise to a height of

Options

A. 35 m
B. 65 m
C. 100 m
D. 135 m

Answer

To solve the problem step by step, we will follow the principles of energy conservation and the effects of inelastic collisions.

### Step 1: Calculate the initial potential energy (PE) of the ball
The ball is dropped from a height of 100 meters. The potential energy (PE) at this height can be calculated using the formula:
\[ \text{PE} = mgh \]
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 (100 m)

### Step 2: Calculate the kinetic energy (KE) just before impact
When the ball is dropped, it converts its potential energy into kinetic energy just before it hits the ground. Therefore, the kinetic energy just before impact is equal to the potential energy at the height of 100 m:
\[ \text{KE} = mgh = mg \cdot 100 \]

### Step 3: Determine the kinetic energy after the impact
The ball loses 35% of its energy upon impact. Therefore, the kinetic energy after the impact (KE') can be calculated as:
\[ \text{KE}' = \text{KE} \times (1 - 0.35) = \text{KE} \times 0.65 \]
Substituting the value of KE:
\[ \text{KE}' = mg \cdot 100 \cdot 0.65 \]

### Step 4: Calculate the maximum height reached after the impact
After the impact, the ball will rise to a new height (h'). The potential energy at this new height can be expressed as:
\[ \text{PE}' = mgh' \]
Setting the potential energy equal to the kinetic energy after the impact:
\[ mgh' = \text{KE}' \]
Substituting the expression for KE':
\[ mgh' = mg \cdot 100 \cdot 0.65 \]

### Step 5: Cancel out the mass (m) and gravitational acceleration (g)
Since \( m \) and \( g \) are present on both sides of the equation, they can be canceled out:
\[ h' = 100 \cdot 0.65 \]

### Step 6: Calculate the final height
Now, calculate \( h' \):
\[ h' = 65 \, \text{meters} \]

Thus, the ball will rise to a height of **65 meters** after the impact.

### Final Answer:
The ball will rise to a height of **65 meters**.

---