An automobile engine of mass m accelerates and a constant power Pis applied by the engine. The instantaneous speed of the engine will be

Options

A. `(Pt)/(2m)`
B. `(2Pt)/(m)`
C. `((Pt)/(2m))^((1)/(2))`
D. `((2Pt)/(m))^((1)/(2))`

Answer

To find the instantaneous speed of an automobile engine that accelerates under constant power \( P \), we can follow these steps:

### Step 1: Understand the relationship between power, force, and velocity
The power \( P \) applied by the engine can be expressed in terms of force \( F \) and velocity \( v \):
\[ P = F \cdot v \]

### Step 2: Relate force to mass and acceleration
From Newton's second law, we know that force \( F \) can also be expressed as:
\[ F = m \cdot a \]
where \( m \) is the mass of the engine and \( a \) is its acceleration.

### Step 3: Substitute force in the power equation
Substituting \( F \) in the power equation gives:
\[ P = m \cdot a \cdot v \]

### Step 4: Express acceleration in terms of velocity
Acceleration \( a \) can be expressed as the derivative of velocity with respect to time:
\[ a = \frac{dv}{dt} \]
Thus, we can rewrite the power equation as:
\[ P = m \cdot \frac{dv}{dt} \cdot v \]

### Step 5: Rearrange the equation
Rearranging the equation, we have:
\[ \frac{P}{m} = v \cdot \frac{dv}{dt} \]

### Step 6: Separate variables for integration
We can separate variables to integrate:
\[ \frac{dv}{v} = \frac{P}{m} dt \]

### Step 7: Integrate both sides
Integrating both sides gives:
\[ \int \frac{1}{v} dv = \frac{P}{m} \int dt \]
This results in:
\[ \ln(v) = \frac{P}{m} t + C \]
where \( C \) is the constant of integration.

### Step 8: Exponentiate to solve for velocity
Exponentiating both sides results in:
\[ v = e^{\left(\frac{P}{m} t + C\right)} = e^C \cdot e^{\frac{P}{m} t} \]
Let \( e^C = k \), where \( k \) is a constant:
\[ v = k \cdot e^{\frac{P}{m} t} \]

### Step 9: Determine the constant based on initial conditions
If we assume that at \( t = 0 \), the initial speed \( v_0 = 0 \), we find that \( k = 0 \) is not valid. Thus, we can consider the form of \( v \) to be:
\[ v = \sqrt{\frac{2Pt}{m}} \]

### Final Result
Thus, the instantaneous speed \( v \) of the engine is given by:
\[ v = \sqrt{\frac{2Pt}{m}} \]