Finding Velocity of Car Slowing Down: v_0-c*(t-t_1)^2/2

[fmiren]

Homework Statement

I don't understand the limits of the definite integral in the acceleration problem below.

Relevant Equations

Acceleration and velocity equations

At time t=0 , a car moving along the + x -axis passes through x=0 with a constant velocity of magnitude v0 . At some time later, t₁ , it starts to slow down. The acceleration of the car as a function of time is given by:

a(t)= 0 0≤t≤t₁
-c(t−t1) t₁<t₂

where c is a positive constants in SI units, and t₁<t≤t₂ is the given time interval for which the car is slowing down. Express your answer in terms of v_0 for v0 , t_1 for t1 , t_2 for t2 , and c as needed. What is v(t) , the velocity of the car as a function of time during the time interval t₁<t≤t₂?

To get the velocity I integrate the accelaeration function and get v_0-c*(t_2-t_1)^2/2 since I think these should be the boundaries of the definite integral. Bu the correct answer is v_0-c*(t-t_1)^2/2 and they integrate from t (upper limit) to t1 (lower limit).
Could you please help me to understand it?

 

[BvU]

Your answer would mean that the velocity is a constant. Your expression does not satisfy
$$a = {dv\over dt} = c(t-t_1) $$since it is independent of $t$.
I.e. your $a =0$ !
Do you agree ?

 

[fmiren]

Yes, I see this now, what confuses me is how to choose the limits of definite integral in such kind of problems.

 

[BvU]

The velocity at some moment $t$ is the integral fom some start up to that same $t$ -- so $t$ is the upper limit of the integration$$v(t)- v(t_1) = \int_{t_1}^t a(\tau)\; d\tau $$
(the integrand is just a 'dummy variable' ; I used $\tau$ to designate it).