Understanding Kinetic Energy and Velocity

Let's analyse the relationship between kinetic energy and velocity. The kinetic energy of a body is the energy it possesses due to its motion. It depends on the mass of the body and its velocity.

The Formula for Kinetic Energy

The standard formula used to calculate kinetic energy (\(KE\)) is:

$$KE = \frac{1}{2} m v^2$$

Where:

• \(m\) is the mass of the body.
• \(v\) is the velocity of the body.

This formula shows that kinetic energy is directly proportional to the mass and the square of the velocity. This squared relationship with velocity is very important for this question.

Step-by-Step Calculation

We are given a scenario where the velocity of a body changes. Let's denote the initial conditions with subscript '1' and the final conditions with subscript '2'.

Step 1: Write down the initial kinetic energy.

Let the initial velocity be \(v_1\). The initial kinetic energy (\(KE_1\)) is:

$$KE_1 = \frac{1}{2} m v_1^2$$

Here, \(m\) is the mass of the body, which remains constant throughout the process.

Step 2: Determine the final velocity.

The question states that the velocity of the body becomes two times its initial velocity. So, the final velocity (\(v_2\)) is:

$$v_2 = 2 v_1$$

Step 3: Write down the final kinetic energy.

Using the kinetic energy formula with the final velocity \(v_2\), the final kinetic energy (\(KE_2\)) is:

$$KE_2 = \frac{1}{2} m v_2^2$$

Step 4: Substitute the final velocity in terms of the initial velocity.

Substitute \(v_2 = 2 v_1\) into the equation for \(KE_2\):

$$KE_2 = \frac{1}{2} m (2 v_1)^2$$

Now, square the term \(2 v_1\):

$$KE_2 = \frac{1}{2} m (4 v_1^2)$$

Step 5: Rearrange the equation to relate final kinetic energy to initial kinetic energy.

We can factor out the number 4 from the equation:

$$KE_2 = 4 \times \left(\frac{1}{2} m v_1^2\right)$$

Notice that the expression inside the parentheses, \(\frac{1}{2} m v_1^2\), is exactly the formula for the initial kinetic energy, \(KE_1\).

So, we can write:

$$KE_2 = 4 \times KE_1$$

Step 6: Determine the value of n.

The question states that the final kinetic energy becomes \(n\) times its initial kinetic energy, which means:

$$KE_2 = n \times KE_1$$

Comparing this with our result \(KE_2 = 4 \times KE_1\), we can see that the value of \(n\) is 4.

Conclusion on Kinetic Energy Change

When the velocity of a body is doubled, its kinetic energy becomes four times its initial kinetic energy. This happens because kinetic energy is proportional to the square of the velocity (\(v^2\)). If \(v\) becomes \(2v\), then \(v^2\) becomes \((2v)^2 = 4v^2\).

Revision Table: Kinetic Energy and Velocity

Additional Information on Kinetic Energy

Kinetic energy is a scalar quantity, meaning it only has magnitude and no direction. It is measured in Joules (J) in the SI system.

• Work-Energy Theorem: The change in kinetic energy of an object is equal to the net work done on the object. This is a fundamental concept in physics connecting work and energy.
• Dependence on Mass: Kinetic energy is also directly proportional to the mass of the object. If the mass doubles (and velocity remains constant), the kinetic energy also doubles.
• Examples: A moving car has kinetic energy. A thrown ball has kinetic energy. The wind (moving air) has kinetic energy.

Understanding how kinetic energy scales with velocity and mass is crucial for solving many physics problems related to motion and energy.