Question

The linear momentum of a particle varies with time as, $P=a+bt+c{{t}^{2}}$. Then which of the following is correct?
A. Velocity of particle is inversely proportional to time
B. Displacement of the particle is independent of time
C. Force varies with time in a quadratic manner
D. Force is dependent linearly on time.

Answer 1

Hint: Linear momentum of a particle is the product of the mass and the velocity of the particle. So, velocity of the particle can be found from the above expression. Then we can find the displacement from the velocity as the velocity is the displacement per unit time. Again, we can find the force on the particle as acceleration can be obtained as the differentiation of velocity with respect to time.

Complete answer:
The linear momentum of a particle can be defined as the product of the mass and the velocity of the particle at an instant.
Mathematically we can express it as,
$P=mv$
Where, P is the linear momentum of the particle, m is the mass of the particle and v is the velocity of the particle.
The dependence of linear momentum of the particle on time is given as,

$P=a+bt+c{{t}^{2}}$
So, we can write
$mv=a+bt+c{{t}^{2}}$
So, velocity can be expressed as,
$v=\dfrac{1}{m}\left( a+bt+c{{t}^{2}} \right)$

So, the velocity of the particle depends on the time and is directly proportional to the time.
We can find the displacement of the particle from the velocity.

The velocity of the particle can be expressed in terms of the distance covered in a time as
$\begin{align}
  & v=\dfrac{s}{t} \\
 & s=vt \\
\end{align}$

Distance can be expressed as,
$\begin{align}
  & s=\dfrac{1}{m}\left( a+bt+c{{t}^{2}} \right)t \\
 & s=\dfrac{1}{m}\left( at+b{{t}^{2}}+c{{t}^{3}} \right) \\
\end{align}$
So, the displacement of the particle depends on time.
the force of the particle can be expressed as,
$F=ma$

Again, acceleration can be expressed as,
$\begin{align}
  & a=\dfrac{dv}{dt}=\dfrac{d}{dt}\left\{ \dfrac{1}{m}\left( a+bt+c{{t}^{2}} \right) \right\} \\
 & a=\dfrac{1}{m}\left( b+2ct \right) \\
\end{align}$
So, force can be expressed as,
$F=ma=b+2ct$
So, the force on the particle is linearly dependent on the time.

So, the correct answer is “Option D”.

Note:
Linear momentum of an object is directly proportional to the mass and the velocity of the object. It can sometimes depend on the time and sometimes independent of the time. From the dependency of the linear momentum on time, we can find the dependency of the other parameters on time.