A car weighing 1400 kg is moving at speed of 54 km/h up a hill when the motor stops. If it is just able to read the destination which is at a height of 10 m above the point calculte the work done against friction (negative of the work done by the friction).

Explanation

To solve the problem step by step, we will use the work-energy theorem and the concepts of kinetic energy and gravitational potential energy.

### Step 1: Convert the speed from km/h to m/s
The initial speed of the car is given as 54 km/h. We need to convert this to meters per second (m/s).

\[ \text{Speed in m/s} = \frac{54 \text{ km/h} \times 1000 \text{ m/km}}{3600 \text{ s/h}} = 15 \text{ m/s} \]

**Hint**: Remember that to convert km/h to m/s, you multiply by \(\frac{1000}{3600}\).

### Step 2: Calculate the initial kinetic energy (KE_initial)
The formula for kinetic energy is given by:

\[ KE = \frac{1}{2} mv^2 \]

Where \(m\) is the mass (1400 kg) and \(v\) is the speed (15 m/s).

\[ KE_{\text{initial}} = \frac{1}{2} \times 1400 \text{ kg} \times (15 \text{ m/s})^2 = \frac{1}{2} \times 1400 \times 225 = 157500 \text{ J} \]

**Hint**: Kinetic energy depends on the mass and the square of the velocity.

### Step 3: Calculate the work done against gravity (W_gravity)
The work done against gravity can be calculated using the formula:

\[ W_{\text{gravity}} = mgh \]

Where \(m = 1400 \text{ kg}\), \(g = 9.8 \text{ m/s}^2\), and \(h = 10 \text{ m}\).

\[ W_{\text{gravity}} = 1400 \text{ kg} \times 9.8 \text{ m/s}^2 \times 10 \text{ m} = 137200 \text{ J} \]

**Hint**: Work done against gravity is equal to the weight of the object times the height it is raised.

### Step 4: Apply the work-energy theorem
According to the work-energy theorem:

\[ \text{Work done by friction} + \text{Work done by gravity} = \Delta KE \]

Where \(\Delta KE = KE_{\text{final}} - KE_{\text{initial}}\). Since the car comes to a stop at the height of 10 m, \(KE_{\text{final}} = 0\).

\[ \Delta KE = 0 - 157500 \text{ J} = -157500 \text{ J} \]

### Step 5: Rearranging the equation to find work done by friction
Now we can rearrange the equation to solve for the work done by friction:

\[ \text{Work done by friction} = \Delta KE - W_{\text{gravity}} \]

Substituting the values we calculated:

\[ \text{Work done by friction} = -157500 \text{ J} - 137200 \text{ J} = -294700 \text{ J} \]

### Step 6: Calculate the work done against friction
The work done against friction is the negative of the work done by friction:

\[ \text{Work done against friction} = -(\text{Work done by friction}) = 294700 \text{ J} \]

**Final Answer**: The work done against friction is **294700 J**.