a body of mass 2kg is applied to a vriable force at x=0, f = 4N; at x=1m, f=0 and at x=2m, f= -4 N and then after x=2 and more f remains constant i.e. -4 N. What is KE at x=3 m?

To determine the kinetic energy (KE) of the body at x = 3 m, we need to analyze the work done on the body by the variable force as it moves from its initial position to the final position. The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. Let's break this down step by step.

Understanding the Force Profile

We have a body of mass 2 kg subjected to a variable force that changes with position:

• At x = 0 m, the force (F) is 4 N.
• At x = 1 m, the force drops to 0 N.
• At x = 2 m, the force becomes -4 N.
• For x > 2 m, the force remains constant at -4 N.

Calculating Work Done from 0 m to 1 m

First, we calculate the work done as the body moves from x = 0 m to x = 1 m. The force is linearly decreasing from 4 N to 0 N. The average force over this interval can be calculated as:

Average Force = (F_initial + F_final) / 2 = (4 N + 0 N) / 2 = 2 N

The work done (W) can be calculated using the formula:

W = Average Force × Distance = 2 N × 1 m = 2 J

Calculating Work Done from 1 m to 2 m

Next, we analyze the work done from x = 1 m to x = 2 m. Here, the force changes from 0 N to -4 N. Again, we find the average force:

Average Force = (0 N + (-4 N)) / 2 = -2 N

The work done in this segment is:

W = Average Force × Distance = -2 N × 1 m = -2 J

Calculating Work Done from 2 m to 3 m

Finally, we consider the movement from x = 2 m to x = 3 m, where the force is constant at -4 N:

W = Force × Distance = -4 N × 1 m = -4 J

Total Work Done

Now, we can find the total work done on the body as it moves from x = 0 m to x = 3 m by summing the work done in each segment:

Total Work = Work (0 to 1 m) + Work (1 to 2 m) + Work (2 to 3 m)

Total Work = 2 J + (-2 J) + (-4 J) = 2 J - 2 J - 4 J = -4 J

Relating Work to Kinetic Energy

According to the work-energy principle, the total work done on the body is equal to the change in kinetic energy:

ΔKE = W_total

If we assume the body starts from rest at x = 0 m, its initial kinetic energy (KE_initial) is 0 J. Therefore:

KE_final = KE_initial + W_total = 0 J - 4 J = -4 J

Interpreting the Result

A negative kinetic energy does not have a physical meaning in this context, indicating that the body has lost energy due to the work done against the force. In practical terms, this means that the body has not gained any kinetic energy and may have come to a stop or is moving in the opposite direction due to the negative force acting on it.

In summary, at x = 3 m, the kinetic energy of the body is effectively zero or negative, indicating that it has been decelerated significantly by the applied forces.