The displacement of a body moving along a straight line is givne by `S =bt^(n)` where 'b' is a constant and 't' is time. For what value of 'n' the body moves under the action of constant force ? .

Options

A. `3//2`
B. `1`
C. `2`
D. `1//2`

Explanation

To determine the value of 'n' for which a body moves under the action of a constant force, we start with the given displacement equation:

\[ S = b t^n \]

where \( b \) is a constant and \( t \) is time.

### Step 1: Find the Velocity
The velocity \( v \) of the body is the first derivative of displacement \( S \) with respect to time \( t \):

\[ v = \frac{dS}{dt} \]

Calculating the derivative:

\[ v = \frac{d}{dt}(b t^n) = b \cdot n \cdot t^{n-1} \]

### Step 2: Find the Acceleration
The acceleration \( a \) is the derivative of velocity \( v \) with respect to time \( t \):

\[ a = \frac{dv}{dt} \]

Calculating the derivative of velocity:

\[ a = \frac{d}{dt}(b n t^{n-1}) = b n (n-1) t^{n-2} \]

### Step 3: Analyze the Condition for Constant Force
According to Newton's second law, the force \( F \) acting on the body is given by:

\[ F = m \cdot a \]

For the force to be constant, the acceleration \( a \) must also be constant. This means that the expression for acceleration should not depend on time \( t \).

From the acceleration equation:

\[ a = b n (n-1) t^{n-2} \]

For \( a \) to be constant, the exponent of \( t \) must be zero:

\[ n - 2 = 0 \]

### Step 4: Solve for 'n'
Solving the equation:

\[ n - 2 = 0 \implies n = 2 \]

### Conclusion
Thus, for the body to move under the action of a constant force, the value of \( n \) must be:

\[ n = 2 \]