A body of mass m has its position x at a time t, expressed by the equation :
`x=3t^(3//2)+2t-(1)/(2)`. The instantaneous force F on the body is proportional to

Options

A. `t^(3//2)`
B. t
C. `t^(-1//2)`
D. `t^(0)`

Answer

To solve the problem step by step, we need to find the instantaneous force \( F \) on the body, which is proportional to the acceleration. The position \( x \) of the body is given by the equation:

\[ x = 3t^{3/2} + 2t - \frac{1}{2} \]

### Step 1: Calculate the Velocity
The first step is to find the velocity \( v \) of the body, which is the first derivative of the position \( x \) with respect to time \( t \):

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

Differentiating the position function:

\[ v = \frac{d}{dt}(3t^{3/2} + 2t - \frac{1}{2}) \]

Using the power rule for differentiation:

\[ v = 3 \cdot \frac{3}{2} t^{3/2 - 1} + 2 \cdot 1 \]

\[ v = \frac{9}{2} t^{1/2} + 2 \]

### Step 2: Calculate the Acceleration
Next, we find the acceleration \( a \), which is the derivative of the velocity \( v \):

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

Differentiating the velocity function:

\[ a = \frac{d}{dt}\left(\frac{9}{2} t^{1/2} + 2\right) \]

Using the power rule again:

\[ a = \frac{9}{2} \cdot \frac{1}{2} t^{1/2 - 1} + 0 \]

\[ a = \frac{9}{4} t^{-1/2} \]

### Step 3: Relate Force to Acceleration
The instantaneous force \( F \) on the body is given by Newton's second law:

\[ F = m \cdot a \]

Substituting the expression for acceleration:

\[ F = m \cdot \frac{9}{4} t^{-1/2} \]

### Step 4: Determine the Proportionality
Since \( m \cdot \frac{9}{4} \) is a constant, we can say that the force \( F \) is proportional to \( t^{-1/2} \):

\[ F \propto t^{-1/2} \]

### Conclusion
The instantaneous force \( F \) on the body is proportional to \( t^{-1/2} \).

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