Answer to Question #350174 in Calculus for HappyFeet

Question

Derive an equation x(t) for the instantaneous position of the car as a function of time. Identify the

●      value x when t = 0 s

●      asymptote of this function as t → ∞

t(0-28 m/s) (s) t(400m)(s) t (Maxspeed)(s)

2.5 9.75 8.0

Answer 1

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

Using the information that $t_{maxspeed}=8.0\ s$ we have

$$v(t) = A (1-e ^ {- t/8.0})$$

Using the information that $t (0-28 m/s)$ is $2.5\ s$ we have

$$v(2.5) = A (1-e ^ {- 2.5/8.0})=28$$ $$A=\dfrac{28}{1-e ^ {- 2.5/8.0}}m/s$$

$$A=104.328\ m/s$$

$$v(t) = 104.328 (1-e ^ {- t/8.0})$$

$$x(t)=\int 104.328 (1-e ^ {- t/8.0})dt$$

$$=104.328 (t+8e ^ {- t/8.0})+C$$

$$x(9.75)=104.328 (9.75+8e ^ {- 9.75/8.0})+C=400$$

$$C=-863.912\ m$$

$$x(t)=104.328 (t+8e ^ {- t/8.0})-863.912$$

$$x(0)=104.328(8)-863.912$$

$$x(0)=-29.288\ m$$

$$x(t)\to\infin\ as\ t\to \infin$$