A body of mass `2 kg` is thrown up vertically with kinetic energy of `490 J`. If `g = 9.8m//s^(2)`, the height at which the kinetic energy of the body becomes half of the original value, is

Options

A. (a)50 m
B. (b)25 m
C. (c)12.5 m
D. (d)19.6 m

Answer

To solve the problem, we need to find the height at which the kinetic energy of a body becomes half of its original value. Let's break down the solution step by step.

### Step 1: Identify the given values
- Mass of the body (m) = 2 kg
- Initial kinetic energy (KE_initial) = 490 J
- Acceleration due to gravity (g) = 9.8 m/s²

### Step 2: Calculate the final kinetic energy
Since we need to find the height at which the kinetic energy becomes half of its original value, we calculate:
\[ KE_{final} = \frac{KE_{initial}}{2} = \frac{490 \, \text{J}}{2} = 245 \, \text{J} \]

### Step 3: Apply the conservation of energy principle
According to the conservation of energy:
\[ KE_{initial} + PE_{initial} = KE_{final} + PE_{final} \]
Initially, the potential energy (PE_initial) is 0 because the body is at the ground level (height = 0). Therefore, we can simplify the equation:
\[ KE_{initial} = KE_{final} + PE_{final} \]
Substituting the known values:
\[ 490 \, \text{J} = 245 \, \text{J} + PE_{final} \]

### Step 4: Calculate the potential energy at the final state
Rearranging the equation to find the potential energy:
\[ PE_{final} = 490 \, \text{J} - 245 \, \text{J} = 245 \, \text{J} \]

### Step 5: Use the formula for potential energy
The potential energy (PE) at height (h) is given by:
\[ PE = m \cdot g \cdot h \]
Substituting the values we have:
\[ 245 \, \text{J} = 2 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 \cdot h \]

### Step 6: Solve for height (h)
Rearranging the equation to solve for h:
\[ h = \frac{245 \, \text{J}}{2 \, \text{kg} \cdot 9.8 \, \text{m/s}^2} \]
Calculating the denominator:
\[ h = \frac{245}{19.6} \]

### Step 7: Perform the final calculation
Calculating the height:
\[ h = 12.5 \, \text{m} \]

### Conclusion
The height at which the kinetic energy of the body becomes half of the original value is **12.5 meters**.

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