Question

An object of mass ${\text{ 100 kg }}$is accelerated uniformly from a velocity of ${\text{ 5 m}}{{\text{s}}^{ - 1}}{\text{ }}$to ${\text{ 8 m}}{{\text{s}}^{ - 1}}$ in ${\text{ 6 s}}{\text{. }}$Calculate the initial and final momentum of the object. Also, find the magnitude of the force exerted on the object.

Answer 1

According to Newton's second law of motion, the rate of change of momentum of a body is directly proportional to the applied force and the change takes place in the direction of the force. According to this law, the force is proportional to the rate of change of momentum. The momentum of a body can be defined as the product of mass and velocity.
Formula used
$P = mv$ (Where, ${\text{ }}P{\text{ }}$stands for the momentum of the body, ${\text{ m }}$stands for the mass of the body, and ${\text{ v }}$stands for the velocity of the body)
$F = \dfrac{{mv - mu}}{t}$ (Where, ${\text{ F }}$stands for the force acting on an object ${\text{ mv }}$stands for the final momentum of the body, ${\text{ mu }}$stands for the initial momentum of the body)

Step by step solution
Using second law, we have to find the initial and final momentum of the body. We know that the momentum of a body is the product of mass and velocity. It is given by,
$P = mv$
The mass of the body is given by,
$m = 100{\text{ }}kg$
The initial velocity is given by,
$u = 5{\text{ m/s}}$
Therefore, the initial momentum can be written as,
$P = mu = 100 \times 5 = 500{\text{ kg m/s}}$
The final velocity of the body can be written as,
$v = 8{\text{ }}m/s$
Therefore, the final momentum of the body can be written as,
$P = mv = 100 \times 8 = 800{\text{ kg m/s}}$
Now we have the initial and final momentum. We have to find the magnitude of force exerted on the object.
Force is proportional to the change in momentum. The force acting on the body is the rate of change of momentum with time. That is given by,
$F = \dfrac{{mv - mu}}{t}$
The initial momentum, ${\text{ mu = 500 kg m/s}}$
The final momentum, ${\text{ mv = 800 kg m/s}}$
The time taken for the change in velocity is given by, ${\text{ t = 6 s}}$
Substituting these values in the expression for force,
$F = \dfrac{{800 - 500}}{6} = 50N$

Note
Another method to find the force acting on the body is given below.
We know that force can also be given as the product of mass and acceleration.
$F = ma$(Where, ${\text{ F }}$stands for the force acting on an object, ${\text{ m }}$stands for the mass of the object and ${\text{ a }}$stands for the acceleration of the object)
Acceleration is the rate of change of velocity with time,
$a = \dfrac{{v - u}}{t}$
The initial velocity ${\text{ u = 5 m/s}}$
The final velocity ${\text{ v = 8 m/s }}$
The time taken for the velocity change, ${\text{ }}t = 6s$
Then, the acceleration is,
$a = \dfrac{{8 - 5}}{6} = 0.5{\text{ m/}}{{\text{s}}^2}$
Now we can find the force as,
$F = ma = 100 \times 0.5 = 50N$