Acceleration and Velocity Problem

[DeadxBunny]

Could someone please explain to me how to do the following problem?:

The acceleration of an object is given by a(t) = (-6t, 12t-32, 0). If the initial velocity is v(0) = (1,0,1) and the position at t=1 is known to be r(1) = (2,0,1), find r(t).

I know that to find velocity when given acceleration, I should integrate acceleration and to find r(t) I should integrate whatever equations I get for velocity. But, how do I use the v(0) = (1,0,1) and r(1) = (2,0,1) stuff?

 

[cepheid]

Those are your inital conditions...remember that when you integrate the a(t) function, the most general antiderivative will be the answer you get plus an arbitrary constant C (because the C disappears when you differentiate v(t) again to get a(t)). What this means is that you have an infinite number of solutions for v(t) Physically, the C represents your initial velocity in this case. Can you see that no matter what your initial velocity is, the rate of change will be the same...so any number of v(t) functions each with a different constant term representing the initial velocity will satisfy the equation (they all have the same accelaration..their velocities all change by the same amount over a specified time interval regardless of what they were to start with). Fortunately the initial velocity is conveniently provided for you in the problem so that you can determine a specific solution for v(t). Same thing goes for when you integrate v(t) to get r(t).

 

[DeadxBunny]

Ok I think I got it now. Thank you so much!