A Car Has an Engine Which Delivers a Constant Power
Introduction
A car has an engine which delivers a constant power. The car is moving on a level road and the resistance to motion is constant. The mass of the car is m and the coefficient of resistance is μ. The car starts from rest.
Derivation of Equations of Motion
The equation of motion for the car is:
$$ma = P – μmg$$
where:
* $$m$$ is the mass of the car
* $$a$$ is the acceleration of the car
* $$P$$ is the power delivered by the engine
* $$mu$$ is the coefficient of resistance
* $$g$$ is the acceleration due to gravity
Since the power is constant, the acceleration of the car is constant. Therefore, we can integrate the equation of motion to obtain the velocity of the car as a function of time:
$$v = frac{P}{mmu}t + C$$
where $$C$$ is a constant of integration. To determine the value of $$C$$, we use the initial condition that the car starts from rest:
$$v(0) = 0$$
Substituting this into the equation for velocity, we get:
$$C = 0$$
Therefore, the equation for velocity becomes:
$$v = frac{P}{mmu}t$$
We can also integrate the equation of motion to obtain the displacement of the car as a function of time:
$$s = frac{P}{2mmu}t^2 + D$$
where $$D$$ is a constant of integration. To determine the value of $$D$$, we use the initial condition that the car starts from rest:
$$s(0) = 0$$
Substituting this into the equation for displacement, we get:
$$D = 0$$
Therefore, the equation for displacement becomes:
$$s = frac{P}{2mmu}t^2$$
Discussion
The equations of motion for a car with a constant power engine show that the acceleration, velocity, and displacement of the car are all proportional to the power delivered by the engine. This means that the more powerful the engine, the faster the car will accelerate, move, and cover a certain distance.
The equations of motion also show that the acceleration, velocity, and displacement of the car are all inversely proportional to the mass of the car. This means that the heavier the car, the slower it will accelerate, move, and cover a certain distance.
Finally, the equations of motion show that the acceleration, velocity, and displacement of the car are all inversely proportional to the coefficient of resistance. This means that the greater the resistance to motion, the slower the car will accelerate, move, and cover a certain distance.
