Question

Calculate the change in kinetic energy of a moving body if its velocity is reduced to $\frac{1}{3}$of the initial velocity.

Answer 1

Step 1: Assumptions

Let,

The mass of the body be $\left(m\right)$

Initial velocity be $\left(v\right)$

As the velocity is reduced to $\frac{1}{3}$of the initial velocity, then

Final velocity be $\left(\frac{v}{3}\right)$

Let initial kinetic energy be $\left(K_1\right)$

And the final kinetic energy be $\left(K_2\right)$

Step 2: Formula used

According to the formula of kinetic energy

$$\begin{gathered} KIneticenergy=\frac{1}{2}\times mass\times {\left(velocity\right)}^2 \\ K=\frac{1}{2}m{v}^2........\left(a\right) \end{gathered}$$

And the formula of change in energy

$$\begin{gathered} Changeinkineticenergy=Initialkineticenergy-finalkineticenergy \\ ∆K=K_1-K_2.......\left(b\right) \end{gathered}$$

Step 3: Finding the initial kinetic energy

On, putting all the given values in equation (a), we get

$K_1=\frac{1}{2}m{v}^2.......\left(c\right)$

Step 4: Finding the final kinetic energy

On, putting all the given values in equation (a), we get

$$\begin{gathered} K_2=\frac{1}{2}m\times {\left(\frac{v}{3}\right)}^2 \\ {K}_2=\frac{1}{2}\times m\times \frac{v^2}{9} \\ {K}_2=\frac{1}{9}\times \frac{1}{2}m{v}^2.......\left(d\right) \end{gathered}$$

Step 5: Calculating the change in kinetic energy

On, putting equation (c) and equation (d) in equation (b), we get

$$\begin{gathered} ∆K=\frac{1}{2}m{v}^2-\frac{1}{9}\times \frac{1}{2}m{v}^2 \\ ∆K=\frac{1}{2}m{v}^2\left(1-\frac{1}{9}\right) \\ ∆K=\frac{8}{9}\times \frac{1}{2}m{v}^2 \\ ∆K=\frac{8}{9}\times K_1 \end{gathered}$$

Hence, kinetic energy become $\frac{8}{9}$of initial kinetic energy.