A motorcycle (mass of cycle plus rider = 3.0 102 kg) is traveling at a steady speed of 20 m/s. The force of air resistance acting on the cycle and rider is 1.8 102 N. Find the power necessary to sustain this speed under the following conditions.

(a) the road is level

(b) the road is sloped upward at 35° with respect to the horizontal

To find the power necessary to sustain the speed of the motorcycle under different conditions, we can use the formula:

Power = Force × Velocity

(a) When the road is level, the force required to overcome air resistance is equal to the force of air resistance acting on the cycle and rider. Therefore,

Power = Force of air resistance × Velocity

Power = (1.8 × 10^2 N) × (20 m/s)
Power = 3600 N·m/s or 3600 Watts (W)

So, the power necessary to sustain the speed on level ground is 3600 W.

(b) When the road is sloped upward at 35° with respect to the horizontal, we need to consider the component of the force of gravity that acts parallel to the direction of motion. This component should be balanced by the force of air resistance.

The force of gravity can be calculated using the formula:

Force of gravity = mass × acceleration due to gravity

Force of gravity = (3.0 × 10^2 kg) × (9.8 m/s^2)
Force of gravity = 2940 N

The component of the force of gravity parallel to the direction of motion can be found by multiplying the force of gravity by the cosine of the angle of inclination:

Component of force of gravity = Force of gravity × cos(angle)

Component of force of gravity = 2940 N × cos(35°)
Component of force of gravity = 2410 N

The force required to overcome the force of gravity is equal to the sum of the force of air resistance and the component of force of gravity parallel to the direction of motion:

Force = Force of air resistance + Component of force of gravity

Force = 1.8 × 10^2 N + 2410 N
Force = 2590 N

Now we can calculate the power:

Power = Force × Velocity

Power = 2590 N × 20 m/s
Power = 51,800 N·m/s or 51,800 W

So, the power necessary to sustain the speed on a road sloped upward at 35° with respect to the horizontal is 51,800 W.

a) P=F*v=1.8x10^2N*20m/s

b) P=(F+mg(sin 35))v

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