An engine is working at a constant power draws a load of mass m against a resistance r. Find maximum speed of load and time taken to attain half this speed.

Options

A. `t=(Pm)/(8r^(2))`
B. `t=(Pm)/(8r)`
C. `t=(Pm)/(r^(2))`
D. `t=(Pm)/(9r^(2))`

Answer

To solve the problem of finding the maximum speed of a load being drawn by an engine working at constant power against a resistance, and the time taken to attain half this speed, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Forces Involved**:
The engine is working against a resistance \( R \) while moving a load of mass \( m \). The resistance can be thought of as a force opposing the motion.

2. **Power and Force Relationship**:
The power \( P \) of the engine can be expressed in terms of the force \( F \) and the velocity \( v \):
\[ P = F \cdot v \]
Here, the force \( F \) exerted by the engine must overcome the resistance \( R \):
\[ F = R \]

3. **Setting Up the Equation**:
Since the engine's power is constant, we can relate the power to the resistance and the velocity:
\[ P = R \cdot v \]
Rearranging gives:
\[ v = \frac{P}{R} \]
This equation gives us the maximum speed \( v_{\text{max}} \) of the load when the engine is working at constant power.

4. **Finding the Maximum Speed**:
Thus, the maximum speed \( v_{\text{max}} \) of the load is:
\[ v_{\text{max}} = \frac{P}{R} \]

5. **Calculating Time to Reach Half Maximum Speed**:
To find the time taken to reach half of the maximum speed, we need to consider the relationship between power, force, and acceleration. The acceleration \( a \) of the load can be expressed as:
\[ F = m \cdot a \]
Since \( F = R \), we have:
\[ R = m \cdot a \implies a = \frac{R}{m} \]

6. **Using Kinematics**:
We can use the kinematic equation to find the time \( t \) taken to reach half the maximum speed \( \frac{v_{\text{max}}}{2} \):
\[ v = u + at \]
Here, \( u = 0 \) (initial speed), \( v = \frac{v_{\text{max}}}{2} \), and \( a = \frac{R}{m} \):
\[ \frac{P}{2R} = 0 + \left(\frac{R}{m}\right) t \]
Rearranging gives:
\[ t = \frac{P}{2R} \cdot \frac{m}{R} = \frac{Pm}{2R^2} \]

7. **Final Expression for Time**:
Thus, the time taken to reach half the maximum speed is:
\[ t = \frac{Pm}{2R^2} \]

### Summary of Results:
- Maximum speed \( v_{\text{max}} = \frac{P}{R} \)
- Time to reach half maximum speed \( t = \frac{Pm}{2R^2} \)