Answer to Question #350174 in Calculus for HappyFeet

Question #350174

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



1
Expert's answer
2022-06-13T16:16:06-0400
v(t)=A(1et/tmaxspeed)v(t) = A (1-e ^ {- t/t_{maxspeed}})


Using the information that tmaxspeed=8.0 st_{maxspeed}=8.0\ s we have



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

Using the information that t(028m/s)t (0-28 m/s) is 2.5 s2.5\ s we have



v(2.5)=A(1e2.5/8.0)=28v(2.5) = A (1-e ^ {- 2.5/8.0})=28A=281e2.5/8.0m/sA=\dfrac{28}{1-e ^ {- 2.5/8.0}}m/s




A=104.328 m/sA=104.328\ m/s




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

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

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

x(9.75)=104.328(9.75+8e9.75/8.0)+C=400x(9.75)=104.328 (9.75+8e ^ {- 9.75/8.0})+C=400

C=863.912 mC=-863.912\ m

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


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

x(0)=29.288 mx(0)=-29.288\ m


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

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