A body is projected vertically up with a velocity of 60 m/s. The percentage of its initial kinetic energy converted into potential energy after 3 s is `(g=10ms^(-2))`

Options

A. 0.25
B. 0.75
C. 0.5
D. 0.875

Answer

To solve the problem, we need to find the percentage of the initial kinetic energy that is converted into potential energy after 3 seconds for a body projected vertically upward with an initial velocity of 60 m/s. Given that \( g = 10 \, \text{m/s}^2 \), we can follow these steps:

### 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 \( m \) is the mass of the body and \( v \) is the initial velocity.

Substituting the values:
\[ KE_{\text{initial}} = \frac{1}{2} m (60)^2 = \frac{1}{2} m (3600) = 1800m \, \text{Joules} \]

### Step 2: Calculate the Height (h) after 3 seconds
We can use the second equation of motion to find the height:
\[ s = ut + \frac{1}{2} a t^2 \]
where:
- \( u = 60 \, \text{m/s} \) (initial velocity)
- \( a = -10 \, \text{m/s}^2 \) (acceleration due to gravity, negative because it acts downward)
- \( t = 3 \, \text{s} \)

Substituting the values:
\[ s = (60)(3) + \frac{1}{2} (-10)(3^2) \]
Calculating each term:
\[ s = 180 - \frac{1}{2} \times 10 \times 9 = 180 - 45 = 135 \, \text{meters} \]

### Step 3: Calculate the Potential Energy (PE) at height h
The formula for potential energy is:
\[ PE = mgh \]
Substituting the values:
\[ PE = m \times 10 \times 135 = 1350m \, \text{Joules} \]

### Step 4: Calculate the Percentage of Kinetic Energy Converted to Potential Energy
The percentage of kinetic energy converted to potential energy is given by:
\[ \text{Percentage} = \left( \frac{PE}{KE_{\text{initial}}} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage} = \left( \frac{1350m}{1800m} \right) \times 100 \]
The mass \( m \) cancels out:
\[ \text{Percentage} = \left( \frac{1350}{1800} \right) \times 100 = 0.75 \times 100 = 75\% \]

### Final Answer
The percentage of the initial kinetic energy converted into potential energy after 3 seconds is **75%**.

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