If two bodies of different masses, initially at rest, are acted upon by the same force for the same time, then the both bodies acquire the same

When two bodies of different masses are subjected to the same force for the same duration, they will experience different accelerations due to their differing masses. However, they will acquire the same momentum by the end of the time interval. Let’s break this down step by step to understand how this works.

Understanding Force, Mass, and Acceleration

According to Newton's second law of motion, the relationship between force (F), mass (m), and acceleration (a) is expressed as:

F = m × a

This means that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. If two bodies are acted upon by the same force, the acceleration of each body will differ based on its mass.

Calculating Acceleration

Let’s say we have two bodies: Body A with mass m_A and Body B with mass m_B. If both are acted upon by the same force F for the same time t, we can express their accelerations as follows:

• For Body A: a_A = F / m_A
• For Body B: a_B = F / m_B

From this, it’s clear that if m_A is less than m_B, then a_A will be greater than a_B. This means Body A will accelerate faster than Body B.

Momentum Acquisition

Now, let’s talk about momentum. Momentum (p) is defined as the product of mass and velocity:

p = m × v

When the same force is applied for the same time, both bodies will experience a change in momentum. The impulse experienced by each body, which is the product of force and time, is equal to the change in momentum:

Impulse = F × t = Δp

Applying Impulse to Both Bodies

Since both bodies are subjected to the same force for the same duration, the impulse is the same:

• Impulse for Body A: F × t = Δp_A
• Impulse for Body B: F × t = Δp_B

This means that both bodies will acquire the same change in momentum, regardless of their masses. Therefore, even though they accelerate differently, the total momentum gained by both bodies will be equal.

Example for Clarity

Imagine you have a small cart (Body A) weighing 1 kg and a larger cart (Body B) weighing 3 kg. If you push both carts with a force of 6 N for 2 seconds:

• For Body A: a_A = 6 N / 1 kg = 6 m/s²
• For Body B: a_B = 6 N / 3 kg = 2 m/s²

After 2 seconds, Body A will have a velocity of 12 m/s, while Body B will have a velocity of 4 m/s. However, the momentum for both will be:

• Momentum of Body A: p_A = 1 kg × 12 m/s = 12 kg·m/s
• Momentum of Body B: p_B = 3 kg × 4 m/s = 12 kg·m/s

As you can see, both bodies acquire the same momentum of 12 kg·m/s, even though their masses and velocities differ.

Final Thoughts

This principle illustrates the fundamental relationship between force, mass, acceleration, and momentum. It’s a great example of how physics helps us understand the behavior of objects in motion under the influence of forces. So, in summary, while the two bodies will have different accelerations and velocities, they will indeed acquire the same momentum when acted upon by the same force for the same time period.