A machine rated as 150W, changes the velocity of a 10kg mass from `4 ms^(-1)` to `10ms^(-1)` in 4s. The efficiency of the machine nearly is

Options

A. 0.7
B. 0.3
C. 0.5
D. 0.4

Answer

To find the efficiency of the machine, we will follow these steps:

### Step 1: Calculate the Work Done by the Machine
The work done (W) on the mass can be calculated using the formula for kinetic energy:
\[ W = \Delta KE = \frac{1}{2} m (v_f^2 - v_i^2) \]
Where:
- \( m = 10 \, \text{kg} \) (mass)
- \( v_f = 10 \, \text{m/s} \) (final velocity)
- \( v_i = 4 \, \text{m/s} \) (initial velocity)

Substituting the values:
\[ W = \frac{1}{2} \times 10 \times (10^2 - 4^2) \]
\[ W = \frac{1}{2} \times 10 \times (100 - 16) \]
\[ W = \frac{1}{2} \times 10 \times 84 \]
\[ W = 5 \times 84 = 420 \, \text{Joules} \]

### Step 2: Calculate the Average Power Applied by the Machine
The average power (P) can be calculated using the formula:
\[ P = \frac{W}{t} \]
Where:
- \( W = 420 \, \text{J} \) (work done)
- \( t = 4 \, \text{s} \) (time taken)

Substituting the values:
\[ P = \frac{420}{4} = 105 \, \text{W} \]

### Step 3: Calculate the Efficiency of the Machine
Efficiency (η) can be calculated using the formula:
\[ \eta = \frac{P_{\text{applied}}}{P_{\text{rated}}} \]
Where:
- \( P_{\text{applied}} = 105 \, \text{W} \) (average power)
- \( P_{\text{rated}} = 150 \, \text{W} \) (rated power)

Substituting the values:
\[ \eta = \frac{105}{150} \]
\[ \eta = 0.7 \]

### Step 4: Convert Efficiency to Percentage
To express efficiency as a percentage:
\[ \eta_{\text{percentage}} = \eta \times 100\% \]
\[ \eta_{\text{percentage}} = 0.7 \times 100\% = 70\% \]

### Final Answer
The efficiency of the machine is approximately **70%**.

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