Question

Find the body's position at time t a=10cos 5t, v (0)= -10,s (0)=10

Answer 1

`a(t)=10\cos5t`

`v(t)=\displaystyle\int 10\cos5t\,\mathrm dt=2\sin5t+C_1`

Since
`v(0)=-10`, you have


`-10=2\sin0+C_1\implies C_1=-10`

so


`v(t)=2\sin5t-10`

Next,


`s(t)=\displaystyle\int(2\sin5t-10)\,\mathrm dt=-\frac25\cos5t-10t+C_2`

Since
`s(0)=10`, you get


`10=-\frac25\cos0-10*0+C_2\implies C_2=\frac{52}5`

and so


`s(t)=-\frac25\cos5t-10t+\frac{52}5`