Answer

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**Hint:**Velocity is defined as the rate of change of position of an object with respect to time. And the velocity is also defined as the amount of distance travelled by an object in a given amount of time. By using the velocity formula, the solution can be determined.

**Complete step by step solution**

1. Relation between velocity and kinetic energy

$KE = \dfrac{1}{2} \times m{v^2}$

Where, $KE$ is the kinetic energy, $m$ is the mass, $v$ is the velocity.

$KE = \dfrac{1}{2} \times m{v^2}$

If the velocity is doubled,

$KE = \dfrac{1}{2} \times m{\left( {2v} \right)^2}$

Squaring the terms inside the bracket,

$KE = \dfrac{1}{2} \times m\left( {4{v^2}} \right)$

By arranging the above equation,

$KE = 4 \times \left( {\dfrac{1}{2} \times m{v^2}} \right)$

By this equation, we clearly understand that the velocity is doubled then the kinetic energy becomes 4 times.

2. Relation between velocity and acceleration

Acceleration is the rate of change of velocity with respect to time. If the velocity is doubled, then it is due to acceleration only. In other words, by changing the acceleration, the velocity is doubled. So, if the velocity is doubled, the acceleration will not double.

3. Relation between velocity and momentum

By Linear momentum equation,

$p = m \times v$

Where, $p$ is the momentum, $m$ is the mass, $v$ is the velocity.

$p = m \times v$

As the velocity is doubled,

$p = m \times \left( {2v} \right)$

By arranging the above equation,

$p = 2\left( {mv} \right)$

From the above equation, it is clear that the velocity is doubled then the momentum also doubled.

4.Relation between velocity and potential energy:

Actually, there is no relationship between velocity and potential energy. If the potential energy is changed to kinetic energy, then there is a relation between velocity and kinetic energy.

**Hence, the option (C) is correct.**

**Note:**The velocity of the object is doubled by changing the acceleration only. If the velocity is doubled its kinetic energy is multiplied by four times. And there is no relationship between the velocity and potential energy. So, if the velocity is doubled, momentum also doubles.

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