Question

If the linear momentum of a sphere is doubled, its kinetic energy will become _________ its original value.
A. Twice
B. Four times
C. Eight times
D. Sixteen times.

Answer 1

Hint: We use the relation between kinetic energy and linear momentum. According to it, kinetic energy is directly proportional to the mass of the object and the square of velocity of the object.

Formula used: $K = \dfrac{1}{2}m{v^2}$
Where, K is the kinetic energy
               m is the mass,
               v is the velocity

Complete step by step answer:
Given, let the initial momentum of sphere be P and kinetic energy be K.
From the problem we can it is given as initial momentum is doubled then final momentum will be ${P_1} = 2P$ then
We are interested to find the final kinetic energy, which is ${K_1} = ?$
Now, Let us find the initial momentum $P = mv$ ………… (1)
We can consider the Initial kinetic energy $K = \dfrac{1}{2}m{v^2}$ ………….(2)
Then from equation (1), we know that the momentum is given as the product of mass and velocity that is,$P = mv$
Let us take squaring on both the sides, we get
${P^2} = {m^2}{v^2} = m \times m{v^2}$
⇒$m{v^2} = \dfrac{{{P^2}}}{m}$
Now we can substitute the value of $m{v^2}$ in equation (2) we get,
⇒$K = \dfrac{1}{2}\dfrac{{{P^2}}}{m} = \dfrac{{{P^2}}}{{2m}}$
This is the relation between kinetic energy and momentum.
Now, we can calculate the final kinetic energy,
⇒${K_1} = \dfrac{{{P_1}^2}}{{2m}} = \dfrac{{{{\left( {2P} \right)}^2}}}{{2m}}$
⇒${K_1} = 4 \times \dfrac{{{P^2}}}{{2m}} = 4K$
Therefore, from the calculations we found that the final kinetic energy will be 4 times the original kinetic energy.

Hence, the correct option is option(B).

Additional information:
It is common to experience that a moving body has the capacity to impart motion to those with which it collides. The effect of a collision will depend both on the mass of the moving body and its velocity. For example, even though a bicycle and a motor car may be traveling with the same velocity, the effect of collision with a given body will be much greater in the case of a motor car than in the case of the bicycle. Similarly, if two motor cars of equal mass are moving with different velocities, the effect of collision with a given body will be greater in the case of the car moving faster. This property of the moving body which depends on its mass and velocity was called the quantity of motion by Newton and it is termed as momentum.

Note:
The momentum of a moving body is defined as the capacity of the body imparting velocity to another body and it is measured as the product of the body and its velocity. That is p=mv.
In physics, kinetic energy is the energy acquired by an object when it is moving.
It is typically found out that, the total energy is comprised of potential energy and kinetic energy that is:
Total energy = Kinetic energy + Potential Energy