Choosing Limits of Integration for Velocity Calculation

[fmiren]

Homework Statement

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, t[SUB]1[/SUB] , it starts to slow down. The acceleration of the car as a function of time is given by:

a(t)= 0 0≤t≤t[SUB]1[/SUB]
-c(t−t1) t[SUB]1[/SUB]<t[SUB]2[/SUB]

where c is a positive constants in SI units, and t[SUB]1[/SUB]<t≤t[SUB]2[/SUB] 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 t[SUB]1[/SUB]<t≤t[SUB]2[/SUB]?

Relevant Equations

Acceleration and velocity equations

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?

 

[jbriggs444]

Your formula for $v(t)$ does not depend on t. But velocity during the interval from $t_1$ to $t_2$ changes over time. So it must depend on t.

Suppose that $t$ is equal to $t_1$. We know that at this time, $v(t) = v_0$. What does your formula give as an answer?

 

[fmiren]

If t = t1, and v(t) = v_0, then v_0-c(t-t_1)^2/2 = v_0 which means -c(t-t_1) should be 0? Am I right?

 

[jbriggs444]

If t = t1, and v(t) = v_0, then v_0-c(t-t_1)^2/2 = v_0 which means -c(t-t_1) should be 0? Am I right?

Except for the fact that your formula does not contain $-c(t-t_1)$. Instead it contains $-c(t_2-t_1)$.

 

[fmiren]

Aa, sorry. Ok, I am trying to get it, But then how do I choose the limits of integral? I'm confused. If the problem says that t<t1<t2, I think the natural (which seems natural to me) is t2 and t1. This part confuses me.

 

[jbriggs444]

Aa, sorry. Ok, I am trying to get it, But then how do I choose the limits of integral? I'm confused. If the problem says that t<t1<t2, I think the natural (which seems natural to me) is t2 and t1. This part confuses me.

One thing to take care of before we get started is notation. We never want to evaluate an integral and use the same variable name for both the dummy variable inside the integral and as variable name in the limits of the integral.

Wrong:$$\int_{t_0}^{t} f(t)\ dt$$Right (change the variable name in the limit):$$\int_{t_0}^{t_\text{now}} f(t)\ dt$$Right (change the dummy variable name):$$\int_{t_0}^t f(x)\ dx$$The dummy variable name is irrelevant. It is just a placeholder. You are free to call it anything you like. Except that you should not re-use another variable name -- that's just begging for confusion.

Now then. How do you choose the limits of integration? You want to integrate the incremental changes in velocity so that you can figure out the total change in velocity. You know the velocity at $t_1$ and you want to know the velocity at $t$. So you integrate from $t_1$ to $t$.