Answer to Question #105088 in Calculus for Leon

Question #105088
Given data is that car has a model that states use equation (7 at top 0 and bottom of integral) 7^0∫v(t)dt, 0-28m/s of 2.6s, time of reach 400m of 10.46s and tmaxspeed of 7s. I have calculated that value of coefficient A is 90.25m/s and time to reach max acceleration is 23.7m/s². I'm certain that the max velocity is at t=7 v(7)=90.25 x (1-e^-10 = 57.05m/s). Please draw a graph for velocity v time showing velocity m/s on y axis also labelled with A, max velocity/speed. Then on the x axis please show where tmax is hit and where t400 is hit with x axis labelled time (s) please also show Vmax along the graph. once this is done please work out the areas for both sections and write a brief explanatation on the comparison between the model and my results :)
1
Expert's answer
2020-03-11T14:21:55-0400
v(t)=90.25(1et/7)v(t)=90.25(1-e^{-t/7})v(2.6)=90.25(1e2.6/7)28(m/s)v(2.6)=90.25(1-e^{-2.6/7})\approx28 (m/s)v(7)=90.25(1e7/7)57(m/s)v(7)=90.25(1-e^{-7/7})\approx57 (m/s)
s=09.675v(t)dt=09.67590.25(1et/7)dt=s=\displaystyle\int_{0}^{9.675}v(t)dt=\displaystyle\int_{0}^{9.675}90.25(1-e^{-t/7})dt==90.25[t+7et/7]9.6750=90.25(9.675+7e9.675/707)=90.25\bigg[t+7e^{-t/7}\bigg]\begin{matrix} 9.675\\ 0 \end{matrix}=90.25(9.675+7e^{-9.675/7}-0-7)\approx=400.01(m)=400.01 (m)

v(9.675)=90.25(1e9.675/7)67.59(m/s)v(9.675)=90.25(1-e^{-9.675/7})\approx67.59 (m/s)

rectangle:67.59 m/s× 9.675 s=653.93 mrectangle: 67.59\ m/s \times\ 9.675 \ s=653.93\ m

Check the work of the model and compare results


v(2.6)=90.25(1e2.6/7)28(m/s), Truev(2.6)=90.25(1-e^{-2.6/7})\approx28 (m/s),\ True

v(7)=90.25(1e7/7)57(m/s)v(7)=90.25(1-e^{-7/7})\approx57 (m/s)

Time of reach 400m


t400=9.675 s<10.46 st_{400}=9.675 \ s<10.46 \ s

δ=9.67510.4610.46100%7.5%\delta={|9.675-10.46| \over 10.46}\cdot100\%\approx 7.5\%

I think based on the results, a model can be adopted to solve the problem.



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