# Simple dynamics problem, I can't seem to get the answer to

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## Homework Statement

A car travels around a circular track having a radius of 300m such that when it is at point A, it has a velocity f 5m/s, which is increasing at the rate of $\dot{v}=0.06t m/s^2$. Determine the magnitudes of the velocity and acceleration when it has traveled one-third the way around the track

## Homework Equations

n,t-coordinate system

$$a=\sqrt{a_t^2+a_n^2}$$

$$a_n= \frac{v^2}{r}$$

## The Attempt at a Solution

Since the radius r=300m, the total distance the car will travel is $2 \pi r= 600\pi m$

So I want to find v and a when the distance = 200pi

Now at A, $\dot{v}=a_t=0.06t$
Initially at A,t=0 and v=5

so

$$\int^{v} _{5} = \int^{t} _{0} 0.06t dt$$
$$v=5+0.03t^2$$

Thus
$$\int ^{s} _{0}= \int ^{t} _{0} (5+0.03t^2) dt$$

$$s=5t+0.01t^3$$

When $s=200\pi$

$$200\pi=5t+0.01t^3$$

Which I do not know how to solve since there is no rational root.

Was I going on the correct track?

Doc Al
Mentor
$$200\pi=5t+0.01t^3$$

Which I do not know how to solve since there is no rational root.
There is a real solution. (Yes, you're on the right track.)

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There is a real solution. (Yes, you're on the right track.)

Well I had to use an online calculator to get t=35.579993691668676.

But my normal calculator doesn't have a function for cubic equations. How would I normally solve it without a computer? I know the rational root theorem but if I were to use the newton-raphson method, I'd spend a lot of time finding a starting point and doing the iteration.

Doc Al
Mentor
Solving a general cubic equation by hand is a bear. There's an analytic solution, but I wouldn't dare attempt it from memory. (I too have been spoiled by fancy calculators.) Here's one version: http://mathworld.wolfram.com/CubicFormula.html" [Broken]

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