# Calculus Problem - Maximum Velocity, Derivatives, etc.

1. Nov 27, 2007

### demersal

[SOLVED] Calculus Problem - Maximum Velocity, Derivatives, etc.

1. The problem statement, all variables and given/known data
A particle moves along a line so that at any time t its position given by x(t)=2$$\Pi$$t + cos2$$\Pi$$t.

What is the maximum velocity?

2. Relevant equations
We found:
v(t) = 2$$\Pi$$ - sin2t$$\Pi$$(2$$\Pi$$)
a(t) = 2$$\Pi$$(-cos2t*$$\Pi$$)(2$$\Pi$$)
all values of t when particle's at rest in [0,3]: t=1/4, 5/4, 9/4

3. The attempt at a solution

We tried setting the acceleration to zero and got t = 1/4, and plugged that in to the velocity and got v(t) = 0, which makes no sense because the max velocity is not when it is at rest.

Any help would be GREATLY appreciated ... I have been working at this for 6 hours and am afraid that I am slowly withering away

Last edited: Nov 27, 2007
2. Nov 27, 2007

### Avodyne

a(t) is NOT zero for t=1/4. You made a mistake.

If the cosine of something is zero, what is the sine? (This is the quick way to do this problem ...)

3. Nov 27, 2007

### hotcommodity

How is a(1/4) not zero? You'd have:

$$a(1/4) = -4 \pi^2 cos(2(1/4) \pi)$$

The cosine of pi over 2 is zero.

The thing is that he's missing values for when a(t) is zero. t should have values of 1/4, 3/4, 5/4, 7/4, and 9/4. Some of these values, when plugged into the velocity equation, do not amount to zero velocity.

4. Nov 27, 2007

### demersal

thank you hotcommodity! i see that now! you are a lifesaver, truly.