MHB Distance Travelled at Time t: Solve Velocity Problem w/ a=5, b=4, c=1

[dave dx]

the velocity of an object is given by v=a+b exp(-ct)-2t
given the values of a, b and c below the distance travled at time t is...?

a=5 b=4 c=1

I have been banging my head against the wall for over a week over this. Any help or tips that can be given would be appreciated.

 

[MarkFL]

To find the distance $D$ traveled, we must use:

$$D(t)=\int_{t_i}^{t_f}\left|v(t)\right|\,dt$$

So, we need to find any roots of the given velocity function:

$$v(t)=5+4\exp(-t)-2t$$

We will need to use a numeric root finding technique (such as Newton-Raphson) to approximate the root(s). Using such a method, I find one root at:

$$t\approx2.6423828566319685138$$

So, if we call this root $r$ (and change the dummy variable of integration, letting $t_i=0,\,t_f=t$), we may write:

$$D(t)=\begin{cases}\displaystyle \int_0^t 5+4e^{-u}-2u\,du & t\le r \\[3pt] \displaystyle \int_0^r 5+4e^{-u}-2u\, du-\int_r^t 5+4e^{-u}-2u\,du, & r<t \\ \end{cases}$$

Can you proceed?

 

[dave dx]

Many thanks for getting back to me. Alas I had gotten the root before and am not yet up to speed enough to understand why you've substituted "u" in?

I have evaluated the equation from 0 to the root shown to the answer 9.94493 and aparantly this is incorrect. so I am obviosly still making mistakes somewhere.

Also i assume that the changing of 5 to a 4 was a typo? (Otherwise I have severally misunderstood the calculus that I have learned)

One final point, as I am also unsure on terminology/ layout of university level mathematics, I assume that your two final equations are presented as different ways to achieve the same answer and as such do not depend on one another?

I am sorry for the inconvenience but I am trying to learn a lot at a fast rate.

 

[MarkFL]

Many thanks for getting back to me. Alas I had gotten the root before and am not yet up to speed enough to understand why you've substituted "u" in?

When evaluating definite integrals, the variable of integration is said to be a "dummy" because it gets "integrated out." It is considered bad form to have the same variable in the inner integral (integrand and differential) as in the limits, and so that's why I changed.

I have evaluated the equation from 0 to the root shown to the answer 9.94493 and aparantly this is incorrect. so I am obviosly still making mistakes somewhere.

I used W|A and got $9.94496$, so you seem to have done it correctly. You are being asked for the general function I assume, and what you did was find the distance traveled up to the point at which the velocity is zero. So, what you did was correct, but not what the problem is actually asking.

W|A - integral of 5+4e^(-x)-2x from 0 to 2.6423828566319685138

Also i assume that the changing of 5 to a 4 was a typo? (Otherwise I have severally misunderstood the calculus that I have learned)

Yes, that was a typo, which I have fixed...

One final point, as I am also unsure on terminology/ layout of university level mathematics, I assume that your two final equations are presented as different ways to achieve the same answer and as such do not depend on one another?

I am sorry for the inconvenience but I am trying to learn a lot at a fast rate.

The function I presented (in which you are to evaluate the given definite integrals) is a piecewise-defined function. Which piece you should use depends on the value of $t$. If $t\le r$, then you use the first piece, and if $r<t$, then you use the second part.

Asking questions here is NOT an inconvenience. Our (MHB's) primary reason for being is to help people with mathematics. :D