A train crusing at a constant velocity Vt related to the ground. A person initially at rest(relative to the train) starts running and gains relative speed V0 after a duration Dt. if the person would have started from rest on to the ground, would the magnitude of impulse exerted on the person during Dt would be smaller ,equal ,or larger than the impulse needed to cause the same change in relative velocity in the same interval on the moving train ? assume that the person motion is in the direction of motion of train.

To tackle this question, we need to break down the concepts of impulse, velocity, and relative motion. Impulse is defined as the change in momentum of an object when a force is applied over a period of time. It can also be understood as the product of the average force applied and the time duration over which it is applied. In this scenario, we have a person running inside a train that is moving at a constant velocity, and we want to compare the impulse experienced by the person when they start running relative to the train versus if they had started running from rest on the ground.

Understanding Impulse and Momentum

Impulse (J) can be mathematically expressed as:

• J = F * Dt
• J = Δp = m * Δv

Where:

• F is the average force applied
• Dt is the time duration
• Δp is the change in momentum
• m is the mass of the person
• Δv is the change in velocity

Scenario Analysis

Let’s analyze the two scenarios:

1. Person Running Inside the Train

In this case, the person starts from rest relative to the train and gains a speed V0 after a time Dt. The change in velocity for the person is simply V0, as they are moving in the same direction as the train. The impulse experienced by the person can be calculated as:

• J_train = m * V0

2. Person Running from Rest on the Ground

If the same person started running from rest on the ground, they would need to overcome the train's velocity (Vt) in addition to achieving the speed V0. Therefore, the total change in velocity for the person would be Vt + V0. The impulse in this scenario would be:

• J_ground = m * (Vt + V0)

Comparing the Impulses

Now, let’s compare the two impulses:

• Impulse inside the train: J_train = m * V0
• Impulse on the ground: J_ground = m * (Vt + V0)

From this, we can see that:

• J_ground = m * Vt + m * V0

Clearly, J_ground is greater than J_train because it includes the additional term m * Vt, which accounts for the train's velocity. Therefore, the impulse exerted on the person when starting from rest on the ground is larger than the impulse needed to achieve the same change in relative velocity inside the moving train.

Conclusion

In summary, the magnitude of impulse exerted on the person during the time Dt when starting from rest on the ground is larger than the impulse needed to cause the same change in relative velocity while running inside the moving train. This difference arises because the person on the ground must not only reach their running speed but also match the train's velocity, resulting in a greater overall change in momentum.