Question

You are working as a Junior Engineer for a small motor racing team. You have been given a proposed mathematical model to calculate the velocity of a car accelerating from rest in a straight line. The equation is: 𝑣(𝑡) = 𝐴 (1 − 𝑒 − 𝑡 𝑡𝑚𝑎𝑥𝑠𝑝𝑒𝑒𝑑)

Ascari A10 5.0 V8 - [2006] t=2.8 tm(400)= 10.36 tmaxspeed= 7.8

derive an equation a(t) for the instantaneous position of the car as a function of time? Identify the acceleration of the car at t = 0s and asymptote of this function as t → ∞?

Answer 1

$$v(t) = A (1-e ^ {- t/t_{maxspeed}})$$

Using the information that $t_{maxspeed}=7.8\ s$ we have

$$v(t) = A (1-e ^ {- t/7.8})$$

Using the information that $t (0-28 m/s)$ is $2.8\ s$ we have

$$v(2.8) = A (1-e ^ {- 2.8/7.8})=28$$ $$A=\dfrac{28}{1-e ^ {- 2.8/7.8}}m/s$$ $$A=92.836\ m/s$$

$$v_{max}=v(t_{maxspeed})=A (1-e ^ {- t_{maxspeed}/t_{maxspeed}})$$ $$v_{max}=A(1-e^{-1})$$ $$v_{max}=92.836\ m/s\cdot(1-e^{-1})$$ $$v_{max}=58.673\ m/s$$

$$a(t)=\dfrac{dv}{dt}=\dfrac{A}{t_{maxspeed}} (e ^ {- t/t_{maxspeed}})$$ $$=\dfrac{92.836\ m/s}{7.8\ s} (e ^ {- t/7.8}), \ 0\ s\le t\le7.8\ s$$

The function $a(t)$ decreases for $0\ s\le t\le7.8\ s.$

$$a_{max}=a(0)=\dfrac{92.836\ m/s}{7.8\ s}$$

$$a_{max}=11.902\ m/s^2$$

If $t\to\infin,a\to0\ m/s^2$