How do you find the velocity and position vectors if you are given that the acceleration vector is $a(t)= (-4 cos (-2t))i + (-4sin (-2t))j + (-2t)k$ and the initial velocity is $v(0)=i+k$ and the initial position vector is $r(0)= i+j+k$ ?

Question

How do you find the velocity and position vectors if you are given that the acceleration vector is $a(t)= (-4 cos (-2t))i + (-4sin (-2t))j + (-2t)k$ and the initial velocity is $v(0)=i+k$ and the initial position vector is $r(0)= i+j+k$ ?

1 Answer

Impact of this question

26172 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-03-30T01:55:01

First, note that $cos(-2t)=cos(2t)$ and $sin(-2t)=-sin(2t)$ since cosine is an even function and sine is an odd function.

Now integrate to get $v(t)=-2sin(2t)i-2cos(2t)j-t^2k+\vec{c}$ , where $\vec{c}$ is a constant of integration. Since $v(0)=i+k$ and since $sin(0)=0$ and $cos(0)=1$ , it follows that $\vec{c}=i+2j+k$ so that $v(t)=(-2sin(2t)+1)i+(-2cos(2t)+2)j+(1-t^2)k$ .

Now integrate again to get $r(t)=(cos(2t)+t)i+(-sin(2t)+2t)j+(t-\frac{1}{3}t^{3})k+\vec{c}$ . Since $r(0)=i+j+k$ , it follows that $\vec{c}=j+k$ so that $r(t)=(cos(2t)+t)i+(-sin(2t)+2t+1)j+(t-\frac{1}{3}t^{3}+1)k$

Science

Math

Humanities

... and beyond

How do you find the velocity and position vectors if you are given that the acceleration vector is $a(t)= (-4 cos (-2t))i + (-4sin (-2t))j + (-2t)k$ and the initial velocity is $v(0)=i+k$ and the initial position vector is $r(0)= i+j+k$ ?

1 Answer

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you find the velocity and position vectors if you are given that the acceleration vector is a(t)= (-4 cos (-2t))i + (-4sin (-2t))j + (-2t)ka(t)= (-4 cos (-2t))i + (-4sin (-2t))j + (-2t)k and the initial velocity is v(0)=i+kv(0)=i+k and the initial position vector is r(0)= i+j+kr(0)= i+j+k?

1 Answer

Related questions

Impact of this question

How do you find the velocity and position vectors if you are given that the acceleration vector is $a(t)= (-4 cos (-2t))i + (-4sin (-2t))j + (-2t)k$ and the initial velocity is $v(0)=i+k$ and the initial position vector is $r(0)= i+j+k$ ?