Conservation of Linear Momentum: A Case Study of a Moving Trolley and a Walking Man

The concept of conservation of linear momentum is fundamental in understanding the dynamics of moving objects. This principle states that the total linear momentum of a closed system remains constant if no external forces act on it. In this article, we will explore a real-world scenario involving a trolley and a walking man to demonstrate the application of this principle.

Introduction to the Scenario

A trolley of mass 20 kg is initially moving with a velocity of 4 m/s. A man of mass 40 kg walks on the trolley with a velocity of 2 m/s relative to the trolley. Using the principle of conservation of linear momentum, we will determine the final velocity of the trolley after the man walks.

Using Conservation of Linear Momentum to Solve the Problem

Let's break down the problem step-by-step using the principle of conservation of linear momentum.

Step 1: Calculate the Initial Momentum of the System

The initial momentum of the system (trolley man) can be calculated as follows:

[ text{Initial momentum} (text{Mass of trolley} times text{Velocity of trolley}) (text{Mass of man} times text{Relative velocity of man with respect to ground}) ]

Given:

- Mass of trolley ((m_t)) 20 kg - Initial velocity of trolley ((v_t)) 4 m/s - Mass of man ((m_m)) 40 kg - Velocity of man relative to the trolley ((v_{mt})) 2 m/s

First, we need to find the velocity of the man relative to the ground:

[ v_m v_t - v_{mt} 4 text{ m/s} - 2 text{ m/s} 2 text{ m/s} ]

Thus, the initial momentum is:

[ p_{text{initial}} (20 text{ kg} times 4 text{ m/s}) (40 text{ kg} times 2 text{ m/s}) 80 text{ kg m/s} 80 text{ kg m/s} 160 text{ kg m/s} ]

Step 2: Determine the Final Momentum of the System

The final momentum of the system can be expressed as:

[ p_{text{final}} (text{Total mass of the system} times text{Final velocity of the system}) ]

The total mass of the system is the sum of the mass of the trolley and the mass of the man:

[ text{Total mass} m_t m_m 20 text{ kg} 40 text{ kg} 60 text{ kg} ]

Let (v_f) be the final velocity of the trolley (and the system as a whole). Therefore:

[ p_{text{final}} 60 text{ kg} times v_f ]

Step 3: Apply the Principle of Conservation of Linear Momentum

According to the principle of conservation of linear momentum:

[ p_{text{initial}} p_{text{final}} ]

Thus:

[ 160 text{ kg m/s} 60 text{ kg} times v_f ]

Solving for (v_f):

[ v_f frac{160 text{ kg m/s}}{60 text{ kg}} approx 2.67 text{ m/s} ]

Therefore, the final velocity of the trolley is approximately 2.67 m/s.

Verification and Conclusion

In the scenario where the man walks with respect to the trolley, we can also verify the final velocity using a different approach. During the man's walk, the man's velocity relative to the ground changes, and we must recalculate the final momentum.

Step 1: Recalculate the Initial Momentum

Given the same initial conditions:

[ p_{text{initial}} (20 text{ kg} times 4 text{ m/s}) (40 text{ kg} times 2 text{ m/s}) 80 text{ kg m/s} 80 text{ kg m/s} 240 text{ kg m/s} ]

Step 2: Find the Final Velocity of the Man Relative to the Ground

[ v_m v_t - v_{mt} 4 text{ m/s} - 2 text{ m/s} 6 text{ m/s} ]

Step 3: Determine the Final Momentum of the System

The final momentum of the system is:

[ p_{text{final}} (text{Total mass of the system} times text{Final velocity of the system}) ]

Let (v_f) be the final velocity of the trolley. Therefore:

[ p_{text{final}} (20 text{ kg} 40 text{ kg}) times v_f 60 text{ kg} times v_f ]

Using the principle of conservation of linear momentum:

[ p_{text{initial}} p_{text{final}} ]

[ 240 text{ kg m/s} 60 text{ kg} times v_f ]