Acceleration of a particle moving along x-axis at time t, given by `f = f_(0) (2 - t/T)`, where `f_0` and T are constants. Initial velocity is zero. In the time interval between t = 0, and the instant when f = 0, the particle's velocity was found to be `N/M f_(0)T`. The value of `N/M` is ______

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given acceleration function
The acceleration of the particle is given by:
\[ f = f_0 \left(2 - \frac{t}{T}\right) \]
where \( f_0 \) and \( T \) are constants.

### Step 2: Determine when the acceleration becomes zero
To find the time when the acceleration \( f \) becomes zero, we set:
\[ f = 0 \]
This gives us:
\[ 0 = f_0 \left(2 - \frac{t}{T}\right) \]
Solving for \( t \):
\[ 2 - \frac{t}{T} = 0 \]
\[ \frac{t}{T} = 2 \]
\[ t = 2T \]

### Step 3: Set up the equation for velocity
The velocity \( v \) of the particle can be found by integrating the acceleration function. We know that:
\[ a = \frac{dv}{dt} \]
Thus,
\[ dv = f \, dt \]
Substituting the expression for \( f \):
\[ dv = f_0 \left(2 - \frac{t}{T}\right) dt \]

### Step 4: Integrate to find the velocity
We will integrate from \( t = 0 \) to \( t = 2T \):
\[ v = \int_0^{2T} f_0 \left(2 - \frac{t}{T}\right) dt \]

### Step 5: Perform the integration
Breaking the integral into two parts:
\[ v = f_0 \left( \int_0^{2T} 2 \, dt - \int_0^{2T} \frac{t}{T} \, dt \right) \]

Calculating the first integral:
\[ \int_0^{2T} 2 \, dt = 2 \times (2T - 0) = 4T \]

Calculating the second integral:
\[ \int_0^{2T} \frac{t}{T} \, dt = \frac{1}{T} \cdot \left[\frac{t^2}{2}\right]_0^{2T} = \frac{1}{T} \cdot \left(\frac{(2T)^2}{2} - 0\right) = \frac{1}{T} \cdot \frac{4T^2}{2} = \frac{2T^2}{T} = 2T \]

### Step 6: Combine the results
Now substituting back into the equation for \( v \):
\[ v = f_0 \left(4T - 2T\right) = f_0 \cdot 2T \]

### Step 7: Relate the velocity to the given expression
We are given that the velocity is also expressed as:
\[ v = \frac{N}{M} f_0 T \]
Setting the two expressions for \( v \) equal:
\[ f_0 \cdot 2T = \frac{N}{M} f_0 T \]

### Step 8: Solve for \( \frac{N}{M} \)
Dividing both sides by \( f_0 T \) (assuming \( f_0 \) and \( T \) are not zero):
\[ 2 = \frac{N}{M} \]

### Final Answer
Thus, the value of \( \frac{N}{M} \) is:
\[ \frac{N}{M} = 2 \]

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