A motorist driving a 1000-kg car wishes to increase her speed from 20 m/s to 30 m/s in 5 s. Determine the horse-power required to accomplish this increase. Neglect friction .

Options

A. 70hp
B. 90hp
C. 30hp
D. 80hp

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the acceleration
The formula for acceleration \( a \) is given by:

\[ a = \frac{v_f - v_i}{t} \]

Where:
- \( v_f = 30 \, \text{m/s} \) (final velocity)
- \( v_i = 20 \, \text{m/s} \) (initial velocity)
- \( t = 5 \, \text{s} \) (time)

Substituting the values:

\[ a = \frac{30 \, \text{m/s} - 20 \, \text{m/s}}{5 \, \text{s}} = \frac{10 \, \text{m/s}}{5 \, \text{s}} = 2 \, \text{m/s}^2 \]

### Step 2: Calculate the force
Using Newton's second law, the force \( F \) can be calculated as:

\[ F = m \cdot a \]

Where:
- \( m = 1000 \, \text{kg} \) (mass of the car)
- \( a = 2 \, \text{m/s}^2 \) (acceleration)

Substituting the values:

\[ F = 1000 \, \text{kg} \cdot 2 \, \text{m/s}^2 = 2000 \, \text{N} \]

### Step 3: Calculate the displacement
Using the kinematic equation:

\[ v_f^2 = v_i^2 + 2as \]

Rearranging for displacement \( s \):

\[ s = \frac{v_f^2 - v_i^2}{2a} \]

Substituting the values:

\[ s = \frac{(30 \, \text{m/s})^2 - (20 \, \text{m/s})^2}{2 \cdot 2 \, \text{m/s}^2} \]

Calculating:

\[ s = \frac{900 - 400}{4} = \frac{500}{4} = 125 \, \text{m} \]

### Step 4: Calculate the work done
Work \( W \) is given by:

\[ W = F \cdot s \]

Substituting the values:

\[ W = 2000 \, \text{N} \cdot 125 \, \text{m} = 250000 \, \text{J} \]

### Step 5: Calculate the power
Power \( P \) is given by:

\[ P = \frac{W}{t} \]

Substituting the values:

\[ P = \frac{250000 \, \text{J}}{5 \, \text{s}} = 50000 \, \text{W} \]

### Step 6: Convert power to horsepower
To convert watts to horsepower, use the conversion factor \( 1 \, \text{hp} = 746 \, \text{W} \):

\[ \text{Horsepower} = \frac{50000 \, \text{W}}{746 \, \text{W/hp}} \approx 67.07 \, \text{hp} \]

### Final Answer
The horsepower required to accomplish this increase in speed is approximately **67 hp**.

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