Find expression in terms of time for a particle's velocity?

[Smeato24]

Homework Statement

A particle moves along a straight line so that its acceleration t seconds after passing a fixed point O on the line is (2 - 2t) cm/s². Three seconds after passing O, the particle has a velocity of 5 cm/s. Find and expression, in terms of t, for the velocity of the particle after t seconds.

Homework Equations

a = dv/dt = d²x/dt²

The Attempt at a Solution

From the information, a = -2t + 2 when x = 0, and therefore, by anti-differentiating, v = -t² + 2t + c.
I assume that the acceleration of the particle stays at -2t + 2 cm/s² for the rest of its motion, and so tried substituting (t + 3) in for t to get: -2t - 4, the acceleration of the particle after 3 seconds. Here I got stuck. Do I set up simultaneous equations and eliminate? Have I done my anti-differentiation wrong?

Any help appreciated,
Smeato

 

[Ray Vickson]

Homework Statement

A particle moves along a straight line so that its acceleration t seconds after passing a fixed point O on the line is (2 - 2t) cm/s². Three seconds after passing O, the particle has a velocity of 5 cm/s. Find and expression, in terms of t, for the velocity of the particle after t seconds.

Homework Equations

a = dv/dt = d²x/dt²

The Attempt at a Solution

From the information, a = -2t + 2 when x = 0, and therefore, by anti-differentiating, v = -t² + 2t + c.
I assume that the acceleration of the particle stays at -2t + 2 cm/s² for the rest of its motion, and so tried substituting (t + 3) in for t to get: -2t - 4, the acceleration of the particle after 3 seconds. Here I got stuck. Do I set up simultaneous equations and eliminate? Have I done my anti-differentiation wrong?

Any help appreciated,
Smeato

You have a correct antderivative, but from there onwards your logic is wrong. Go back and read again exactly what the question tells you.

 

[Smeato24]

Thanks Ray, it really was a simple question!
"t seconds after passing O" means that you start counting at zero at the Origin :)

The correct answer is: v = -t² + 2t + 8