Calculating Kinetic Energy of Translation

The question asks for the kinetic energy of translation of a body with mass 'm' that attains a velocity 'v' from rest in time 't'.

Kinetic energy is the energy an object possesses due to its motion. For translational motion (movement from one point to another), the kinetic energy depends on the mass of the object and its velocity.

Understanding Translational Kinetic Energy

Translational kinetic energy is given by the formula:

$\text{KE} = \frac{1}{2}mv^2$

This can also be written in decimal form as:

$\text{KE} = 0.5mv^2$

Where:

• $m$ represents the mass of the body
• $v$ represents the velocity of the body

The time 't' taken to reach the velocity 'v' is given in the problem, but it is not needed to calculate the kinetic energy if the final velocity 'v' is already known.

Applying the Formula to the Problem

Given:

• Mass of the body = $m$
• Final velocity of the body = $v$
• Initial velocity = 0 (since it starts from rest)

Using the formula for kinetic energy with the given mass 'm' and velocity 'v':

$\text{KE} = 0.5 \times \text{mass} \times (\text{velocity})^2$

Substituting the given values:

$\text{KE} = 0.5 \times m \times v^2$

$\text{KE} = 0.5mv^2$

Comparing with the Options

Let's examine the provided options:

• Option 1: $mv^2$ - This is double the correct kinetic energy formula.
• Option 2: $mgv^2$ - This includes the acceleration due to gravity 'g', which is related to potential energy or work done by gravity, not purely kinetic energy.
• Option 3: $0.5 mv^2$ - This matches the standard formula for kinetic energy.
• Option 4: $0.5 mgv^2$ - This also incorrectly includes 'g'.

Conclusion on Kinetic Energy Calculation

The kinetic energy of translation for a body of mass 'm' attaining a velocity 'v' is found using the formula $0.5mv^2$. This calculation aligns with Option 3.