What is the work to be done to increase the velocity of a car from `30 km h^(-1)` to `60 km h^(-1)` if the mass of the car is `1500 kg` ?

Options

A. `156200J`
B. `156250J`
C. `156275J`
D. `156300J`

Explanation

To solve the problem of calculating the work done to increase the velocity of a car from \(30 \, \text{km/h}\) to \(60 \, \text{km/h}\) with a mass of \(1500 \, \text{kg}\), we will follow these steps:

### Step 1: Convert velocities from km/h to m/s
To convert the velocities from kilometers per hour to meters per second, we use the conversion factor \( \frac{1000 \, \text{m}}{3600 \, \text{s}} \).

- **Initial Velocity (u)**:
\[ u = 30 \, \text{km/h} = 30 \times \frac{1000}{3600} = \frac{30 \times 1000}{3600} = \frac{25}{3} \, \text{m/s} \]

- **Final Velocity (v)**:
\[ v = 60 \, \text{km/h} = 60 \times \frac{1000}{3600} = \frac{60 \times 1000}{3600} = \frac{50}{3} \, \text{m/s} \]

### Step 2: Calculate the initial kinetic energy (KE_initial)
The formula for kinetic energy is given by:
\[ KE = \frac{1}{2} mv^2 \]
Substituting the mass and initial velocity:
\[ KE_{\text{initial}} = \frac{1}{2} \times 1500 \, \text{kg} \times \left(\frac{25}{3} \, \text{m/s}\right)^2 \]
Calculating \(KE_{\text{initial}}\):
\[ KE_{\text{initial}} = \frac{1}{2} \times 1500 \times \frac{625}{9} = \frac{1500 \times 625}{18} = \frac{937500}{18} = 52083.33 \, \text{J} \]

### Step 3: Calculate the final kinetic energy (KE_final)
Now, we calculate the final kinetic energy using the final velocity:
\[ KE_{\text{final}} = \frac{1}{2} \times 1500 \, \text{kg} \times \left(\frac{50}{3} \, \text{m/s}\right)^2 \]
Calculating \(KE_{\text{final}}\):
\[ KE_{\text{final}} = \frac{1}{2} \times 1500 \times \frac{2500}{9} = \frac{1500 \times 2500}{18} = \frac{3750000}{18} = 208333.33 \, \text{J} \]

### Step 4: Calculate the work done (W)
The work done is the change in kinetic energy:
\[ W = KE_{\text{final}} - KE_{\text{initial}} \]
Substituting the values:
\[ W = 208333.33 \, \text{J} - 52083.33 \, \text{J} = 156250 \, \text{J} \]

### Final Answer
The work done to increase the velocity of the car from \(30 \, \text{km/h}\) to \(60 \, \text{km/h}\) is \(156250 \, \text{J}\).

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