A body of mass 2 kg is thrown up vertically with a kinetic energy of 490 J. If the acceleration due to gravity is 9.8 m `s^(-2)`, the height at which the kinetic energy of the body becomes half of the original value is

Options

A. 50 m
B. 25 m
C. 12.5 m
D. 10 m

Explanation

To solve the problem step by step, we will use the principle of conservation of energy.

### Step 1: Understand the given information
- Mass of the body (m) = 2 kg
- Initial kinetic energy (K1) = 490 J
- Acceleration due to gravity (g) = 9.8 m/s²

### Step 2: Determine the final kinetic energy (K2)
Since we need to find the height at which the kinetic energy becomes half of the original value:
- K2 = K1 / 2 = 490 J / 2 = 245 J

### Step 3: Apply the conservation of energy principle
According to the conservation of energy, the total mechanical energy (kinetic + potential) at the start will equal the total mechanical energy when the body reaches the height where K2 is 245 J.

- Initial total energy (E_initial) = K1 + Potential Energy_initial
- Initial Potential Energy (PE_initial) = 0 (at ground level)
- Therefore, E_initial = K1 + 0 = 490 J

At the height (h) where the kinetic energy is K2:
- Final total energy (E_final) = K2 + Potential Energy_final
- E_final = 245 J + PE_final

### Step 4: Set the initial energy equal to the final energy
Using the conservation of energy:
\[ E_{initial} = E_{final} \]
\[ 490 J = 245 J + PE_{final} \]

### Step 5: Solve for Potential Energy_final (PE_final)
Rearranging the equation gives:
\[ PE_{final} = 490 J - 245 J = 245 J \]

### Step 6: Use the formula for potential energy to find height (h)
The formula for potential energy is:
\[ PE = mgh \]
Substituting the known values:
\[ 245 J = 2 kg \cdot 9.8 m/s² \cdot h \]

### Step 7: Solve for height (h)
Rearranging the equation to find h:
\[ h = \frac{245 J}{2 kg \cdot 9.8 m/s²} \]
\[ h = \frac{245}{19.6} \]
\[ h = 12.5 m \]

### 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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