A particle moving along x-axis has acceleration `f`, at time `t`, given by `f = f_0 (1 - (t)/(T))`, where `f_0` and `T` are constant.
The particle at `t = 0` has zero velocity. In the time interval between `t = 0` and the instant when `f = 0`, the particle's velocity `(v_x)` is :

Options

A. `f_(0)T`
B. `(1)/(2)f_(0)T^(2)`
C. `f_(0)T^(2)`
D. `(1)/(2)f_(0)T`

Answer

To solve the problem, we need to find the velocity of the particle as it moves along the x-axis under the given acceleration function. Let's break it down step by step.

### Step 1: Understand the acceleration function
The acceleration \( f \) of the particle is given by:
\[ f = f_0 \left(1 - \frac{t}{T}\right) \]
where \( f_0 \) and \( T \) are constants.

### Step 2: Relate acceleration to velocity
Acceleration is defined as the rate of change of velocity with respect to time:
\[ f = \frac{dv_x}{dt} \]
Substituting the expression for \( f \):
\[ \frac{dv_x}{dt} = f_0 \left(1 - \frac{t}{T}\right) \]

### Step 3: Rearrange the equation for integration
We can rearrange this equation to separate the variables:
\[ dv_x = f_0 \left(1 - \frac{t}{T}\right) dt \]

### Step 4: Integrate both sides
Now, we integrate both sides. The left side will be integrated from 0 to \( v_x \) (the velocity at time \( t \)), and the right side will be integrated from 0 to \( t \):
\[ \int_0^{v_x} dv_x = \int_0^t f_0 \left(1 - \frac{t}{T}\right) dt \]
The left side simplifies to:
\[ v_x = \int_0^t f_0 \left(1 - \frac{t}{T}\right) dt \]

### Step 5: Solve the right side integral
Now we compute the integral on the right side:
\[ \int_0^t f_0 \left(1 - \frac{t}{T}\right) dt = f_0 \left( \int_0^t 1 dt - \frac{1}{T} \int_0^t t dt \right) \]
Calculating the integrals:
1. \(\int_0^t 1 dt = t\)
2. \(\int_0^t t dt = \frac{t^2}{2}\)

Substituting these results back in:
\[ v_x = f_0 \left( t - \frac{1}{T} \cdot \frac{t^2}{2} \right) \]
\[ v_x = f_0 t - \frac{f_0 t^2}{2T} \]

### Step 6: Find the instant when \( f = 0 \)
To find the time when \( f = 0 \):
\[ f_0 \left(1 - \frac{t}{T}\right) = 0 \implies 1 - \frac{t}{T} = 0 \implies t = T \]

### Step 7: Substitute \( t = T \) into the velocity equation
Now, we substitute \( t = T \) into the velocity equation:
\[ v_x = f_0 T - \frac{f_0 T^2}{2T} \]
\[ v_x = f_0 T - \frac{f_0 T}{2} = \frac{f_0 T}{2} \]

### Final Answer
Thus, the velocity of the particle at the instant when \( f = 0 \) is:
\[ v_x = \frac{f_0 T}{2} \]