Question

Which one of the following statements is not true?
(A) The same force of same time causes the same change in momentum for different bodies.
(B) The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts
(C) A greater opposing force is needed to stop a heavy body than a light body in the same time, if they are moving with the same speed.
(D) The greater the change in the momentum in a given time, the lesser is the force that needs to be applied.

Answer 1

In this question, we are asked to find the statement which is not true. In the options given, you can see that only three quantities are to be considered, first is force, second is the time for which the force acts and third is the momentum due to the force. You can analyze all the options by using only Newton’s Laws of motion.

Complete step by step answer:
Newton’s Laws of motion are as follows:
1. A body continues to be in its state (rest or motion) until an external force is applied on the body.
2. The force on a body is equal to the product of mass of the body and its acceleration.
3. There is always an equal and opposite reaction corresponding to an action.
Let us first consider the 2nd law of motion which is mathematically stated as $F = ma$ where $F$ is the force applied, $m$ is the mass of the body and $a$ is the acceleration of the body.
Now, we know that acceleration of a body is defined as the rate of change of velocity with respect to time and mathematically we have, $a = \dfrac{{dv}}{{dt}}$ where $v$ is the velocity of the body.

Now, let us get back to the equation $F = ma$. Replacing $a$ by $\dfrac{{dv}}{{dt}}$, we get, $F = m\dfrac{{dv}}{{dt}}$. As mass is constant, we can take $m$ inside the derivative and we will get $F = \dfrac{{d(mv)}}{{dt}}$. Recall the definition of momentum. It is defined as the product of mass and velocity and is mathematically given as $p = mv$. Substituting this in the equation of force, we have $F = \dfrac{{dp}}{{dt}}$. So, we define force as the rate of change of momentum with respect to time and the change in momentum also known as impulse is given as $dp = Fdt$. In vector form, $d\overrightarrow p = \overrightarrow F dt$.
So, if the same force acts on different bodies for the same interval of time, the change in momentum is equal. As,
$d\overrightarrow p = \overrightarrow F dt \\\ \Rightarrow \dfrac{{d\overrightarrow p }}{{dt}} = \overrightarrow F$,
the rate of change of momentum is proportional to the force applied and also takes place in the direction in which the force acts. So far, option A and B are true.

For the third option, let us consider bodies having masses $M\,\&\,m$ with $M > m$, moving with same velocities $u$. As the bodies are stopped, the final velocity is zero. According to 2nd law of motion, the change in momentum of heavy body $\Delta {p_M}$ is given as $\Delta {p_M} = M(0 - u) = - Mu$ which has to be equal to ${F_M}\Delta t$ where ${F_M}$ is the force required to stop the heavy body. Similarly, the change in momentum of the light body will be $\Delta {p_m} = - mu = {F_m}\Delta t$. Since the time is the same, the force is directly proportional to the masses of the bodies. So greater the mass, greater will be the opposing force. Option C is also true.

By elimination, option D is not true, let us see why. As you know that the change in momentum in a given time $\Delta t$ is given as $\Delta p = F\Delta t$, meaning that it is directly proportional. So greater the change in momentum, greater the force applied has to be.

Therefore, the statement “the greater the change in the momentum in a given time, the lesser is the force that needs to be applied” is not true.Option D is correct.

Note: You should always keep in mind all the three Newton’s Law of motion. Remember that the force is defined as the change in momentum with respect to time only in the case where mass is treated as a constant. The change in momentum due to applied force for a certain period of time is called the impulse.