A body is being moved from rest along a straight line by a machine delivering constant power. The distance covered by body in time t is proportional to

Options

A. `sqrtt`
B. `t^(3//2)`
C. `t^(3//4)`
D. `t^(2)`

Answer

To solve the problem, we need to determine how the distance covered by a body being moved by a machine delivering constant power is related to time. Let's break it down step by step.

### Step 1: Understand the relationship between power, force, and velocity
The power \( P \) delivered by the machine is given by the equation:
\[ P = F \cdot V \]
where \( F \) is the force applied and \( V \) is the velocity of the body. Since the power is constant, we can denote it as \( P = K \) where \( K \) is a constant.

### Step 2: Relate force to acceleration
From Newton's second law, we know that:
\[ F = m \cdot a \]
where \( m \) is the mass of the body and \( a \) is its acceleration. We can express acceleration in terms of velocity:
\[ a = \frac{dv}{dt} \]
Thus, we can rewrite the force as:
\[ F = m \cdot \frac{dv}{dt} \]

### Step 3: Substitute force into the power equation
Substituting \( F \) in the power equation gives us:
\[ K = m \cdot \frac{dv}{dt} \cdot V \]
Since \( V = \frac{ds}{dt} \) (where \( s \) is the distance), we have:
\[ K = m \cdot \frac{dv}{dt} \cdot \frac{ds}{dt} \]

### Step 4: Rearranging the equation
Rearranging the equation, we can express it as:
\[ m \cdot \frac{dv}{dt} = \frac{K}{V} \]
This implies:
\[ F = \frac{K}{V} \]

### Step 5: Integrate to find the relationship between distance and time
We know that:
\[ v = \frac{ds}{dt} \]
Substituting \( v \) in terms of \( t \):
\[ \frac{ds}{dt} = k' \sqrt{t} \]
where \( k' \) is a constant derived from \( K \) and \( m \).

### Step 6: Integrate to find distance
Now, we integrate both sides:
\[ ds = k' \sqrt{t} \, dt \]
Integrating from \( 0 \) to \( s \) and \( 0 \) to \( t \):
\[ s = k' \int_0^t \sqrt{t} \, dt = k' \left[ \frac{2}{3} t^{3/2} \right]_0^t = \frac{2}{3} k' t^{3/2} \]

### Step 7: Conclusion
Thus, we find that:
\[ s \propto t^{3/2} \]
This means the distance covered by the body in time \( t \) is proportional to \( t^{3/2} \).

### Final Answer
The distance covered by the body in time \( t \) is proportional to \( t^{3/2} \).

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