Velocity of a function

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Mooch22

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A particle is moving along the x-axis so that its velocity (v) at any time t, for 0 is less than or equal to t is less than or equal to 16, is given by v(t) = (e^(2sint))-1. At time t = 0, the particle is at the origin.

A. During what intervals of time is the particle moving to the left? **HOW DO YOU FIND THIS?

B. Find the total distance traveled by the particle from t = 0 to t = 4. **HOW DO YOU FIND THIS?

C. Is there any time t, on the given interval, at which the particle returns to the origin? **HOW DO YOU FIND THIS??

Can I get some help, please? I'm not sure where to start on any of these steps. THANKS SO MUCH! :D
 
a. Particle is moving left when v(t)<0.

b. Take the absolute value of the velocity and integrate on the proper interval.

c. It would return to the origin at some time T for which the definite integral, from 0 to T, of v(t)dt is equal to 0. Graph it (on your calculator) and observe.
 
When the particle is moving to the left it has a negative velocity.
Given that v(t) = (e^(2sint))-1
then this will give a negative velocity when (e^(2sint)) is less than 1
This is what you need to solve (find t)

Part (b) do an intergration of the equation and you change from v(t) to s(t) and solve for the range you need.

Part (c) when s=0 you have the origin conditions - solve for this and you have the answer.
 
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