Question

Say whether the following statement is True or False:
\[\text{Force}=\text{mass}\times \text{velocity}\]

Answer 1

Hint: The force is directly proportional to the change in the momentum. The proportionality can be removed by using the proportionality constant. This momentum equals the product of the mass and velocity. Thus, the change in momentum equals the product of mass and the change in velocity. Using this relation between the force, mass and the change in velocity, we will obtain the required result.

Complete step-by-step solution:
The force is directly proportional to the change in the momentum. The mathematical representation of the same is given as follows.
\[F\propto \dfrac{d}{dt}(p)\]
The proportionality can be removed by using the proportionality constant.
\[F=k\dfrac{d}{dt}(p)\]
This proportionality constant can be removed by equating it to unity.
 \[\begin{align}
  & F=(1)\dfrac{d}{dt}(p) \\
 & \therefore F=\dfrac{d}{dt}(p) \\
\end{align}\]
This momentum equals the product of the mass and velocity. The mathematical representation of the same is given as follows.
\[F=\dfrac{d}{dt}(mv)\]
The mass is a constant quantity, whereas, the velocity can be variable. Thus, the differentiation applies to the velocity and not to the mass. Therefore, the change in momentum equals the product of mass and the change in velocity, so, we get,
\[F=m\dfrac{d}{dt}(v)\]
We know that, the change in velocity, or the rate of change of velocity with time gives the acceleration. Thus, we have,
 \[F=m\times a\]
\[\therefore \] The force equals the product of the mass and acceleration, thus, the statement is wrong.

Note: The mass is a constant quantity, whereas, the velocity can be variable. Thus, the differentiation applies to the velocity and not to the mass. Therefore, the change in momentum equals the product of mass and the change in velocity. The change in velocity equals the acceleration.