Final answer:
The power required for a 1400 kg car to climb a 10° hill at a steady 80 km/h is approximately 3434 watts or 3.43 kW.
Step-by-step explanation:
The power required for a car to climb a hill can be calculated using the formula:
P = Mgh / t + 1 / 2ρ C_d A v^3
where:
- P is the power required,
- M is the mass of the car (1400 kg),
- g is the acceleration due to gravity (approximately 9.8 m/s²),
- h is the height of the hill,
- t is the time taken to climb the hill,
- ρ is the air density,
- C_d is the drag coefficient,
- A is the frontal area of the car,
- v is the velocity of the car.
Assuming a level road, the first term on the right side of the equation Mgh / t can be simplified to ( Mg sin(θ) v ), where ( θ ) is the angle of the hill.
Given:
- Mass (M ): 1400 kg,
- Angle of the hill (θ ): 10°,
- Velocity (v): 80 km/h (convert to m/s: ( {80 x 1000} / {3600} ),
- Gravity (g): 9.8 m/s².
Let's calculate:
P = Mg sin(θ) v
P = (1400 kg) x (9.8 m/s^2) x sin(10^\circ) x ({80 x 1000} / {3600}\)
Calculating this expression will give you the power required for the car to climb the hill. Please note that this calculation assumes no air resistance, and the second term of the power equation (related to air resistance) is not considered here.
P ≈ 981 x 0.173 x 22.22
P ≈ 3434 W
So, the power required is approximately 3434 watts, or 3.43 kW.