Question

A particle is falling under gravity. Air resistance on particle is F = \(-\)kv. Find correct option :

• A.
• B.
• C.
• D.

Hint:

In resistive media, look for the "saturation" or "leveling off" effect in graphs. Acceleration decreases over time, so the slope of the v-t graph must decrease.

Answer 1

The Correct Option is B

To solve this problem, we need to analyze the forces acting on the particle as it falls under gravity with air resistance. The air resistance is given as \( F = -kv \), where \( k \) is a positive constant and \( v \) is the velocity of the particle.

Step-by-Step Analysis:

1. When a particle falls under the influence of gravity, the gravitational force acting on it is given by \( F_{\text{gravity}} = mg \), where \( m \) is the mass of the particle and \( g \) is the acceleration due to gravity.
2. The net force acting on the particle will be the difference between the gravitational force and the air resistance. Therefore, according to Newton’s second law, the net force \( F_{\text{net}} \) is: \(F_{\text{net}} = mg - kv\).
3. According to Newton's second law, this net force also equals the product of mass and acceleration \( (ma) \): \(ma = mg - kv\).
4. Since the particle is falling vertically, the vertical acceleration is denoted as \( a \). Solving the equation for acceleration: \(a = g - \frac{k}{m}v\).
5. When the particle reaches terminal velocity, the net force acting on it will be zero, so the acceleration \( a = 0 \). At this point, gravitational force equals the force due to air resistance: \(mg - kv_{\text{terminal}} = 0\), which implies \(v_{\text{terminal}} = \frac{mg}{k}\).

The setup shows that the velocity increases gradually until air resistance balances the gravitational force, leading to terminal velocity. This behavior matches the correct option given:

The correct option illustrates that as the particle falls under gravity, velocity initially increases rapidly and then approaches a constant value—illustrating terminal velocity due to balancing gravitational force and resistive drag force.