**Suppose that a car starts from rest; its engine exerts a constant force (F_a) on it.**
**a) Find the work done by the engine when the car has traveled 4 m.**
The work done by a force is given by:
$$W = overrightarrow{F} cdot overrightarrow{d}$$
where ( overrightarrow{F}) is the force applied and ( overrightarrow{d}) is the displacement of the object.
In this case, the force is constant and in the same direction as displacement, so the dot product simplifies to:
$$ W = F d$$
We are given that the car starts from rest, so its initial velocity is ( v_i = 0 ). The final velocity of the car after traveling 4 m is given by:
$$v_f^2 = v_i^2 + 2ad$$
where (a) is the acceleration of the car. The acceleration can be found using the equation:
$$a = frac{F_a}{m}$$
where (m) is the mass of the car.
Substituting the given values, we get:
$$v_f^2 = 0 + 2 frac{F_a}{m} (4) = frac{8F_a}{m}$$
The displacement is given as (d = 4 m).
Substituting these values into the work equation, we get:
$$ W = F_ad$$
$$ W = F_a(4)$$
**b) Find the power delivered by the engine as a function of time.**
Power is defined as the rate at which work is done, or:
$$P = frac{dW}{dt}$$
where (dW) is the work done and (dt) is the time interval over which the work is done.
In this case, the work done is given by:
$$ W = F_ad$$
and the displacement is given by:
$$d = vt$$
where (v) is the velocity of the car.
Substituting these equations, we get:
$$P = frac{dW}{dt} = frac{d}{dt}(F_avt)$$
$$P = F_avfrac{dv}{dt}$$
The acceleration of the car is constant, so the velocity as a function of time is:
$$v = at$$
Substituting this equation into the power equation, we get:
$$P = F_a(at)frac{d}{dt}(at)$$
$$P = F_a(at)(a)$$
$$P = F_a^2 t$$
**c) Calculate the power delivered at t = 3 s.**
Substituting (t = 3 s) into the power equation, we get:
$$P = F_a^2 (3)$$
**d) If (F_a = 2000) N and (m = 1000) kg, determine the speed of the car after (t = 3) s.**
Using the equation for acceleration:
$$a = frac{F_a}{m}$$
$$a = frac{2000}{1000} = 2 text{ m/s}^2$$
Using the equation for velocity:
$$v = at$$
$$v = 2 (3) = 6 text{ m/s}$$
**e) Calculate the distance traveled by the car after (t = 3) s.**
Using the equation for displacement:
$$d = vt$$
$$d = 6 (3) = 18 text{ m}$$
**f) Draw graphs of position, speed, acceleration, and power as a function of time.**
**Position vs. Time:**
[Image of a graph of position vs. time]
The position-time graph is a straight line with a positive slope, indicating that the car is moving with constant velocity.
**Speed vs. Time:**
[Image of a graph of speed vs. time]
The speed-time graph is a straight line with a positive slope, indicating that the car is accelerating at a constant rate.
**Acceleration vs. Time:**
[Image of a graph of acceleration vs. time]
The acceleration-time graph is a horizontal line, indicating that the car is accelerating at a constant rate.
**Power vs. Time:**
[Image of a graph of power vs. time]
The power-time graph is a parabola, indicating that the power delivered by the engine is increasing at a constant rate.
