Question

A body of mass 5kg is moving with a momentum of 10 kg m/s. A force of 0.2 N acts on it in the direction of motion of the body for 10sec. Find the increase in its kinetic energy.
a) 2.8 J
b) 3.2 J
c) 3.8 J
d) 4.4 J

Answer 1

Hint: Initially the body will possess some initial velocity as it has some momentum. When a force of 0.2 N acts for 10sec the velocity of the body will change as it would accelerate. After 10 sec the body will acquire a final velocity and it will be constant as long as there is no longer a force on the body. Since the kinetic energy of a body is proportional to the velocity, the energy gained will also increase. Hence we can find the initial and the final velocity and since the mass of the object does not change, substitute in the expression for kinetic energy and obtain the increase in energy.

Complete step by step answer:
Let us denote the mass of the body as m, Its initial velocity as v and its final velocity after acceleration as V. It is given in the question that the body has an initial momentum of 10kg m/s. The momentum of a body is mathematically defined as,
$p=mv...(1)$ where m is the mass of the body and v is its velocity of motion. Hence the initial velocity (v)of the body is,
$\begin{array}{rl}& p=mv\\ & 10kgm/s=5kg\left(v\right)\\ & v=2m/s\end{array}$
Hence the initial velocity of the body is 2 m/s. further it is said that the body is subjected to a force of 0.2 N for a duration of 10 sec. By definition the rate of change of momentum of a body is equal to the force acting on the body. This can be mathematically be represented as,
$F={\displaystyle \frac{P_{INITIAL}-P_{FINAL}}{t}}$, in this equation is $P_{INITIAL}$,the initial momentum of the body, $P_{FINAL}$ is the final momentum of the body and t is the time for which the force is exerted on the body. In the above case the initial momentum of the body is given as 10kg m/s hence its final momentum is equal to,
$\begin{array}{rl}& F={\displaystyle \frac{P_{FINAL}-P_{INITIAL}}{t}}\\ & 0.2={\displaystyle \frac{P_{FINAL}-10}{10}}\\ & P_{FINAL}=12kgm/s\end{array}$
Hence the final velocity of the body from equation 1 is,
$\begin{array}{rl}& p=mV\\ & 12=5V\\ & V=2.4m/s\end{array}$
The kinetic energy (K.E) of a body of mass m and velocity v is given by,
$K.E={\displaystyle \frac{1}{2}}m{v}^2\text{ J}$. Therefore the increase in the kinetic energy ($\mathrm{\Delta }E$) of the above body when its velocity changes from v to V is given by,
$\begin{array}{rl}& \mathrm{\Delta }E={\displaystyle \frac{1}{2}}m{V}^2-{\displaystyle \frac{1}{2}}m{v}^2\\ & \mathrm{\Delta }E={\displaystyle \frac{1}{2}}m(V^2-v^2)\\ & \mathrm{\Delta }E={\displaystyle \frac{1}{2}}5\times ({2.4}^2-2^2)\\ & \mathrm{\Delta }E=2.5\times (5.76-4)\\ & \mathrm{\Delta }E=\text{4}\text{.4 J}\end{array}$
Hence the increase in kinetic energy is 4.4 J. hence the correct option is d.

Note:
The units of the quantities in the expression for kinetic energy always have been expressed in SI units. This is because the corresponding unit of energy will be given by joules. One more method of solving the question was to find the acceleration and compute the final velocity using Newton’s kinematic equations.