Determine the minimum energy required to accelerate a 1300kg car from 10 to 60 km/h on an uphill road with a vertical rise of 40m. Assume g = 9.809m/s^2

To determine the minimum energy required to accelerate the car, we need to account for two components:

1. **Kinetic Energy Change** (due to the car's change in speed)
2. **Potential Energy Change** (due to the vertical rise)

### Step 1: Calculate the Change in Kinetic Energy

The formula for kinetic energy (\(KE\)) is:

\[
KE = \frac{1}{2} m v^2
\]

where:
- \( m \) is the mass of the car (1300 kg),
- \( v \) is the velocity in meters per second (m/s).

We need to convert the initial and final speeds from km/h to m/s:

\[
v_1 = 10 \, \text{km/h} = \frac{10 \times 1000}{3600} = 2.78 \, \text{m/s}
\]
\[
v_2 = 60 \, \text{km/h} = \frac{60 \times 1000}{3600} = 16.67 \, \text{m/s}
\]

Now, calculate the change in kinetic energy (\( \Delta KE \)):

\[
\Delta KE = \frac{1}{2} m (v_2^2 - v_1^2)
\]

Substitute the values:

\[
\Delta KE = \frac{1}{2} (1300) \left(16.67^2 - 2.78^2\right)
\]

### Step 2: Calculate the Change in Potential Energy

The potential energy (\(PE\)) is given by:

\[
PE = mgh
\]

where:
- \( m = 1300 \, \text{kg} \),
- \( g = 9.809 \, \text{m/s}^2 \) (gravitational acceleration),
- \( h = 40 \, \text{m} \) (vertical rise).

\[
PE = 1300 \times 9.809 \times 40
\]

### Step 3: Calculate the Total Energy Required

The total energy required is the sum of the change in kinetic energy and the change in potential energy:

\[
E_{\text{total}} = \Delta KE + PE
\]

Let's calculate these values.The minimum energy required to accelerate the car from 10 km/h to 60 km/h on an uphill road with a vertical rise of 40 meters is approximately:

- **Total energy required**: \(685,672 \, \text{J}\) (joules)
- **Change in kinetic energy**: \(175,604 \, \text{J}\)
- **Change in potential energy**: \(510,068 \, \text{J}\)

This total energy accounts for both the increase in speed and the elevation gain.

Would you like more details on any of these steps? Here are some related questions:

1. How does friction affect the total energy required?
2. How would the energy change if the car had a different mass?
3. What would be the effect of a different vertical rise on the energy?
4. How can we calculate the power needed if the time for acceleration is known?
5. How does air resistance influence real-world energy calculations?

**Tip**: The potential energy depends only on the height and mass, while the kinetic energy depends on speed changes.

Determine the minimum energy required to accelerate a 1300kg car from 10 to 60 km/h on an uphill road with a vertical rise of 40m. Assume g = 9.809m/s^2.

Learn how to calculate the minimum energy required to accelerate a 1300kg car from 10 to 60 km/h while climbing a hill with a 40m vertical rise. This problem explores key concepts such as kinetic energy, potential energy, and the work-energy principle. Suitable for advanced high school or college-level physics students.

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