A racing car with a mass of 500.0 kg starts is from rest and completes a quarter mile ( 402 m) race in a time of 5.0 s. The race car's final speed is 130 m/s. Neglecting friction, what average power was needed to produce this final speed ?

Options

A. 140hp
B. 1132hp
C. `8.5xx10^(5)`hp
D. 750hp

Answer

To solve the problem of finding the average power needed for the racing car to reach its final speed, we can follow these steps:

### Step 1: Understand the Problem
We are given:
- Mass of the car (m) = 500 kg
- Initial speed (u) = 0 m/s (the car starts from rest)
- Final speed (v) = 130 m/s
- Distance (d) = 402 m
- Time (t) = 5 s

We need to find the average power (P_avg) required to reach this final speed.

### Step 2: Calculate the Change in Kinetic Energy
The change in kinetic energy (ΔK) can be calculated using the formula:
\[ \Delta K = K_f - K_i \]
where:
- \( K_f = \frac{1}{2} m v^2 \) (final kinetic energy)
- \( K_i = \frac{1}{2} m u^2 \) (initial kinetic energy)

Since the car starts from rest, \( K_i = 0 \).

Substituting the values:
\[ K_f = \frac{1}{2} \times 500 \, \text{kg} \times (130 \, \text{m/s})^2 \]

Calculating \( K_f \):
\[ K_f = \frac{1}{2} \times 500 \times 16900 = 250 \times 16900 = 4225000 \, \text{J} \]

Thus, the change in kinetic energy is:
\[ \Delta K = 4225000 \, \text{J} - 0 = 4225000 \, \text{J} \]

### Step 3: Calculate the Average Power
The average power is given by the formula:
\[ P_{avg} = \frac{\Delta K}{t} \]

Substituting the values:
\[ P_{avg} = \frac{4225000 \, \text{J}}{5 \, \text{s}} = 845000 \, \text{W} \]

### Step 4: Convert Power to Horsepower
To convert watts to horsepower, we use the conversion factor:
\[ 1 \, \text{horsepower} = 746 \, \text{W} \]

Thus,
\[ P_{avg} \text{ (in horsepower)} = \frac{845000 \, \text{W}}{746 \, \text{W/hp}} \approx 1132.5 \, \text{hp} \]

### Final Answer
The average power needed to produce the final speed of the racing car is approximately **1132.5 horsepower**.

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