A 710kg car drives at a constant speed of 23m/s . It is subject to a drag force of 500 N. What power is required from the car's engine to drive the car (a) on level ground? (b) up a hill with a slope of 2.0∘ ?
(a)
F=F(dr)
P=Fv=500•23 =11500 N
(b)
F=F(dr)+mgsinα=500+710•9.8•sin2=500+242.8 =742.8 N
P=Fv=742.8•23=17085 N
To calculate the power required from the car's engine, we can use the formula:
Power = force * velocity
(a) On level ground:
The force required to overcome the drag force is equal to the drag force itself. Therefore, the power required on level ground can be calculated as:
Power = drag force * velocity
Power = 500 N * 23 m/s
Power = 11500 W (or 11.5 kW)
(b) Up a hill with a slope of 2.0∘ :
When driving up a hill, we need to consider the component of the car's weight that acts parallel to the hill's slope. This component is given by:
Force parallel = weight * sin(theta)
where theta represents the angle of the slope.
The force required to overcome the drag force and the force parallel to the slope is:
Total force required = drag force + force parallel
Total force required = 500 N + (weight * sin(theta))
First, let's calculate the force parallel:
Force parallel = 710 kg * 9.8 m/s^2 * sin(2.0∘),
Force parallel = 710 * 9.8 * sin(2.0 * π/180),
Force parallel = 710 * 9.8 * 0.0349,
Force parallel ≈ 242.8973 N
Now, we can calculate the total power required as:
Power = (drag force + force parallel) * velocity,
Power = (500 N + 242.8973 N) * 23 m/s,
Power = 742.8973 N * 23 m/s,
Power ≈ 17083.55 W (or 17.08 kW)
Therefore, the power required from the car's engine to drive the car up a hill with a slope of 2.0∘ is approximately 17083.55 W (or 17.08 kW).
To determine the power required from the car's engine, we need to use the formula:
Power = Force × Velocity
(a) On level ground:
In this case, the car is subject to a drag force but does not have to overcome an inclined surface. Therefore, the total force acting on the car is the drag force alone.
Power = Drag Force × Velocity
= 500 N × 23 m/s
= 11500 N·m/s
= 11500 Watts (since 1 N·m/s = 1 Watt)
Therefore, the power required from the car's engine to drive the car on level ground is 11500 Watts.
(b) Up a hill with a slope of 2.0°:
When driving up an inclined surface, the car not only needs to overcome the drag force but also needs to counteract the force due to the gravitational pull acting downhill. Therefore, we need to calculate the total force acting on the car.
The force acting downhill due to gravity is given by:
Force due to gravity = Mass × Acceleration due to gravity
= 710 kg × 9.8 m/s²
= 6968 N
The force component that needs to be overcome by the car is:
Force to overcome the slope = Force due to gravity × Sin(slope angle)
= 6968 N × Sin(2.0°)
= 242.36 N
The total force acting on the car is the sum of the drag force and the force to overcome the slope:
Total Force = Drag Force + Force to overcome the slope
= 500 N + 242.36 N
= 742.36 N
Now, we can calculate the power required:
Power = Total Force × Velocity
= 742.36 N × 23 m/s
= 17047.28 N·m/s
= 17047.28 Watts
Therefore, the power required from the car's engine to drive the car up a hill with a slope of 2.0° is 17047.28 Watts.