Question

In an imaginary atmosphere, the air exerts a small force F on any particle in the direction of the particle's motion. A particle of mass m projected upwards takes a time \[{t_1}\] in reaching the maximum height and \[{t_2}\] in the return journey to the original point. Then:
A. \[{t_1} < {t_2}\]
B. \[{t_1} > {t_2}\]
C. \[{t_1} = {t_2}\]
D. the relation between \[{t_1}\] and \[{t_2}\] depends on the mass of the particle.

Answer 1

Hint: Gravitational force always acts towards the center of the earth in the opposite direction; if a particle moves in an upward direction, that means for upward motion, gravitational force opposes the motion.

Complete step by step solution:
The force exerted by the air on the particle is in the particle's motion to help the motion to occur. We know that the direction of gravitational force is opposite to the direction of motion when a particle moves upwards. This gravitational force attracts the particle due to which speed of the particle during upward motion is reduced.

When a particle moves downwards, both gravitational force and force exerted by air act in the direction of motion, which means they support the motion. So we can say that the particle's speed in downward motion is greater than the particle's speed in an upward motion.
\[{s_2} > {s_1}\]

We also know that a particle's travelling time is given by the ratio of the distance traveled by it and its speed. Mathematically, we can write:
\[t = \dfrac{d}{s}\]
Here t is the traveling time, d is the distance traveled, and s is the particle's speed.

Write the expression for the traveling time of the particle for upward motion.
\[{t_1} = \dfrac{d}{{{s_1}}}\]……(1)

Write the expression for the traveling time of the given particle for downward motion.
\[{t_2} = \dfrac{d}{{{s_2}}}\]……(2)

On dividing equation (1) and equation (2), we can write:
\[\dfrac{{{t_2}}}{{{t_1}}} = \dfrac{{{s_1}}}{{{s_2}}}\]

We already know that \[{s_2} > {s_1}\] so \[{t_1} > {t_2}\].

Therefore, the particle's time in an upward motion is greater than the time taken in a downward motion. Hence, option (B) is correct.

Note: Do not forget to consider the effect of both the forces that are gravitational force and force exerted by the air on the particle. Also, while establishing the final relationship between \[{t_1}\] and \[{t_2}\] take care of the greater than or less than sign.