A particle is moving as given by the data below:

Question

A particle is moving as given by the data below:

 

v(t)=2sin(t)-5cos(t);   s(0)=0

 

s(t)=_____?

Answer 1

Hi again, Hussein.

 

s(t) is just the antiderivative of v(t).   So "un-derive" v(t).

 

Remember, the deriv of sine is cosine.  The deriv of cos is NEGATIVE sine.   So if you are un-deriving a POS sine, it must have started out a NEG COS.  

 

So the 2sin(t)  un-derives to -2cos(t)

 

and the 5cos(t) un-derives to 5sin(t)

 

and there is always the possibility that the original function had a constant that was lost when you took the derivative.  So we must ALWAYS add a constant into the final answer.

 

s(t) = -2cos(t) - 5sin(t) + C

 

Now we use the fact that s(0) = 0 and find

 

s(0) = -2cos(0) - 5sin(0) + C = 0 

 

s(0) = -2 (1) - 5(0) + C = 0

 

-2 + C = 0,  and C = 2

 

 

So the particular solution is

 

s(t) = -2cos(t) - 5sin(t) + 2