Calculus problem
Two particles move along the x-axis and their positions at time 0

Question

Calculus problem
Two particles move along the x-axis and their positions at time 0<t<2pi are given by x1=cos2t and x2=e^(t-3)/2-0.75. For how many values of t do the two particles have the same velocity?

anonymous · Answer

$$x1=\cos2t x2= e ^{(t-3)/2}-0.75$$

anonymous · Answer

0 less than or equal to t less than or equal to 2pi

agent0smith · Answer

You first have to differentiate both functions to find their velocity (since the rate of change of position gives velocity). Then you have to equate the velocity functions to find when they have the same velocity.

anonymous · Answer

so $$2\cos2t= (e ^{(t-3)/2})*1/2$$

anonymous · Answer

is this right?

agent0smith · Answer

Not quite. It looks like you did not differentiate the cos2t correctly. Cos(u) differentiates to (u')(-sin u).

$$\LARGE x _{1}=\cos2t  $$
$$\LARGE x _{2}= e ^{\frac{ t-3 }{ 2 }}-0.75$$

anonymous · Answer

-2sin(2t) then?

agent0smith · Answer

Correct. Now equate and (try to) solve. I'm not sure you'll be able to do it algebraically, you may need a calculator.

anonymous · Answer

i got 4 as my answer

agent0smith · Answer

4 as the number of times their velocity is equal...?

anonymous · Answer

the number of value t where two particles have the same velocity

agent0smith · Answer

https://www.google.com/search?q=0.5*e%5E((x-3)%2F2)%2C+-2*sin(2x)&aq=f&oq=0.5*e%5E((x-3)%2F2)%2C+-2*sin(2x)&aqs=chrome.0.57j62l2.468j0&sourceid=chrome&ie=UTF-8 Looks like four intersections.

agent0smith · Answer

Did you solve it on a calculator? If you did it by hand... show me how.

anonymous · Answer

i did it on a calculator