Question

If two bodies of equal masses move with uniform velocity of $v$ and $3v$ , what will be the ratio of their kinetic energies?

Answer 1

Hint: Let us know about the uniform velocity. Uniform velocity is defined as a state in which a body travels the same distance in the same amount of time. \[v\] is the average velocity of a body during time \[t\]in the equation (\[d = v{\text{ }}t\]). The body is said to be at uniform velocity if both its magnitude and direction do not change with time.

Complete step by step answer:
The kinetic energy of an object is the energy it has due to its motion in physics. It is the amount of work required to accelerate a body of a given mass from rest to a certain velocity.The body retains its kinetic energy after gaining it during acceleration unless its speed changes.

When the body decelerates from its current speed to a condition of rest, it does the same amount of work.The kinetic energy of a non-rotating object of mass \[m\] travelling at a speed \[\;v\] is $\dfrac{1}{2}m{v^2}$ in classical mechanics. Only when $v$ is significantly less than the speed of light is this a good approximation in relativistic mechanics.

Given: Mass of first body $ = m$, Velocity of the first body $ = v$, Mass of the second body $ = m$ and Velocity of second body $ = 3v$.
$\text{The ratio of kinetic energies of the two bodies} = \dfrac{\text{Kinetic energies of first body}}{\text{Kinetic energies of second body}}$
$\text{The ratio of kinetic energies of the two bodies} = \dfrac{{\dfrac{1}{2}m{v^2}}}{{\dfrac{1}{2}m{{(3v)}^2}}} \\
\therefore \text{The ratio of kinetic energies of the two bodies}= \dfrac{1}{9}$

So we can say that the ratio of the kinetic energies $ = 1:9$.

Note: In this question one should be aware that total kinetic energy is conserved because we have not considered the friction and work done by both the masses. If we include the friction here then there will decrease in the total kinetic energy of both the masses because some part of kinetic energy will be converted to heat loss due to friction.