Question

Resistance to the motion of an automobile consists of road friction, which is almost independent of speed, and air drag, which is proportional to speed-squared. For a certain car with a weight of 12000 N, the total resistant force F is given by F = 300 + 1.8v2, with F in newtons and v in meters per second. Calculate the power (in horsepower) required to accelerate the car at 0.90 m/s2 when the speed is 84 km/h.

Answer 1

Answer:

$\dot W = 4310.924\,W (5\,hp)$

Explanation:

Let assume that vehicle is moving on a horizontal surface and the vehicle mass is 1500 kg. The equation for the power needed to accelerate the car is:

$\dot W = \frac{d}{dt} (\vec P) \bullet \vec v + \vec P \bullet \frac{d}{dt}(\vec v)$

The equivalent engine force needed to accelerate the car is derived from the following equation of equilibrium:

$\Sigma F = P - 300-1.8\cdot v^{2} = (1500\,kg)\cdot (0.90\,\frac{m}{s^{2}} )$

$P = 1850 + 1.8\cdot v^{2}$

The power required at a given velocity is:

$\dot W = 3.6\cdot v^{2} \cdot a + (1850 + 1.8\cdot v^{2})\cdot a$

$\dot W = (1850 + 5.4\cdot v^{2})\cdot a$

The output at $v = 84\,\frac{km}{h}$ is:

$\dot W = [1850 + 5.4\cdot (23.333\,\frac{m}{s})^{2}]\cdot (0.90\,\frac{m}{s^{2}} )$

$\dot W = 4310.924\,W (5\,hp)$