A particle of mass m is at rest at x=0at time t=0 a force given by F=F_oe^-t/Tis applied in the +x direction . Wheret=T, the force is removed. At this moment find

1. Speed of particle
2. Position of particle

To analyze the motion of a particle subjected to a time-dependent force, we can break down the problem into manageable steps. The force acting on the particle is given by \( F = F_0 e^{-t/T} \), where \( F_0 \) is the initial force, \( T \) is a time constant, and \( t \) is the time. Since the particle starts from rest at \( x = 0 \) and \( t = 0 \), we can find both the speed and position of the particle at the moment the force is removed (i.e., at \( t = T \)).

Step 1: Determine the Acceleration

According to Newton's second law, the acceleration \( a \) of the particle can be expressed as:

a = \frac{F}{m}

Substituting the expression for force:

a = \frac{F_0 e^{-t/T}}{m}

Step 2: Find the Velocity

To find the velocity of the particle, we need to integrate the acceleration with respect to time. The velocity \( v(t) \) is given by:

v(t) = \int a \, dt = \int \frac{F_0 e^{-t/T}}{m} \, dt

Carrying out the integration, we have:

• Let \( u = -\frac{t}{T} \), then \( du = -\frac{1}{T} dt \) or \( dt = -T du \).
• The integral becomes:
• v(t) = -\frac{F_0}{m} \int e^{u} (-T) \, du = \frac{F_0 T}{m} e^{-t/T} + C

Since the particle starts from rest, the constant of integration \( C = 0 \). Thus, we have:

v(t) = \frac{F_0 T}{m} (1 - e^{-t/T})

Step 3: Calculate Velocity at \( t = T \)

Now, substituting \( t = T \) into the velocity equation:

v(T) = \frac{F_0 T}{m} (1 - e^{-1})

This gives us the speed of the particle at the moment the force is removed.

Step 4: Find the Position

Next, we need to determine the position of the particle. The position \( x(t) \) can be found by integrating the velocity:

x(t) = \int v(t) \, dt = \int \frac{F_0 T}{m} (1 - e^{-t/T}) \, dt

Carrying out this integration:

• The integral can be split into two parts:
• x(t) = \frac{F_0 T}{m} \left( t + T e^{-t/T} \right) + C'

Again, since the particle starts at \( x = 0 \) when \( t = 0 \), we find that \( C' = 0 \). Thus:

x(t) = \frac{F_0 T}{m} \left( t + T e^{-t/T} \right)

Step 5: Calculate Position at \( t = T \)

Substituting \( t = T \) into the position equation:

x(T) = \frac{F_0 T}{m} \left( T + T e^{-1} \right)

This gives us the position of the particle at the moment the force is removed.

Summary of Results

At the moment the force is removed (when \( t = T \)), we have:

• Speed of the particle: \( v(T) = \frac{F_0 T}{m} (1 - e^{-1}) \)
• Position of the particle: \( x(T) = \frac{F_0 T^2}{m} (1 + e^{-1}) \)

This analysis provides a clear understanding of how the particle behaves under the influence of a time-dependent force. If you have any further questions or need clarification on any part of this process, feel free to ask!