Question

A Porsche $944$ Turbo has a rated engine power of $217$ hp. $30\%$ of the power is lost in the drive train, and $70\%$reaches the wheels. The total mass of the car and driver is $1480kg$, and two-thirds of the weight is over the drive wheels.

a. What is the maximum acceleration of the Porsche on a concrete surface where ${\mu }_s=1.00$? Hint: What force pushes the car forward?

b. If the Porsche accelerates at $a_{max}$, what is its speed when it reaches maximum power output?

c. How long does it take the Porsche to reach the maximum power output?

Short answer

a. The maximum acceleration of Porsche is $6.5vm/s^2$.

b. The speed at the time of maximum power output is $11.7m/s$.

c. The Porsche take$1.8s$to reach maximum output.

Step-by-step solution

Content Introduction

The relation between power and velocity is$$\begin{gathered} P=\overrightarrow{F}.\overrightarrow{v} \\ P=Fv\cos \left(\theta \right) \end{gathered}$$

Explanation (Part a)

The power of Porsche in Watt is

$$\begin{gathered} 217hp=217hp\left(\frac{746W}{1hp}\right) \\ =161,882W \end{gathered}$$

The gravitational force on the car is

$$\begin{gathered} (1480kg)(9.8m/s^2) \\ =14504N \end{gathered}$$

The force on drive wheel is

$$\begin{gathered} F_g=(14504N)\frac{2}{3} \\ =9670N \end{gathered}$$

the static friction of tyres on road is

$$\begin{gathered} F_{max}=f_{s,max} \\ m{a}_{max}=\mu n \\ {a}_{max}=\frac{\mu \left(F_g\right)}{m} \\ {a}_{max}=\frac{1.0(9670N)}{1480kg} \\ {a}_{max}=6.5m/s^2 \end{gathered}$$

Explanation (Part b)

The wheel on use $70\%$of the power generated by the motor. Therefore, the power output is

$$\begin{gathered} v_{max}=\frac{P}{F_g} \\ {v}_{max}=\frac{(161,882W)\left({\displaystyle \frac{70}{100}}\right)}{9670N} \\ {v}_{max}=11.7m/s \end{gathered}$$

Explanation (Part c)

Considering the following kinematics equation,

$$\begin{gathered} v_{max}=v_0+a_{max}(t_{min}-t_0) \\ {v}_{max}=a_{max}\left(t_{min}\right) \\ {t}_{min}=\frac{v_{max}}{a_{max}} \\ {t}_{min}=\frac{11.7m/s}{6.5m/s^2} \\ {t}_{min}=1.8s \end{gathered}$$