Determine acceleration from velocity-position plot

1. Aug 27, 2015

bkw2694

1. The problem statement, all variables and given/known data
The velocity of a particle along s-axis is given as v = 5s^(3/2) , where "s" is in millimeters and v is in millimeters/second. Determine the acceleration when "s" is 2 mm.

So I've been having trouble with differential equations, and I think that's where I'm messing up. My classes haven't really covered velocity-position plots, just velocity-time plots and my book is confusing. The answer to the question is 150 mm/s.

2. Relevant equations
∫vdv = ∫a ds

3. The attempt at a solution

First I used the equation to try to solve it.
∫(0 to v) vdv = ∫(0 to 2) a ds
This gave me (1/2)v^2 = 2a
Then I plugged v = 5s^(3/2) into it and got (1/2)(25s^3) = 2a
Then I plugged in s = 2 and got 100 = 2a
Then I finished solving for a and got a = 50 mm/s, which is incorrect.

2. Aug 27, 2015

ehild

The acceleration is not constant, so its integral with respect to s is not a*s.
It is true (applying chain rule) that a=dv/dt = (dv/ds)(ds/dt)=v dv/ds. Find dv/ds and plug in the formula for v.

3. Aug 27, 2015

bkw2694

Sorry if I'm not following correctly, but are you saying no integral is required for this question?

Solving dv/ds gives me (15/2)s^(1/2). Then multiplying that by the original "v" gives me 5*15/2 (s^2) which after plugging in s = 2 gives me 150mm, the correct answer. Is that the correct setup to this question? And are you saying that I can't use the integrals if acceleration isn't constant?

4. Aug 27, 2015

ehild

Can you integrate the acceleration if you do not know what function of s it is ?

5. Aug 27, 2015

bkw2694

So for me to integrate the acceleration, I need to know a(s)?

6. Aug 27, 2015

insightful

You are being sloppy with your units. Now is the time to break this habit.

7. Aug 27, 2015

ehild

The integral of a function ∫f(x)dx is a new function, the derivative of which is equal to f(x). F(x)= ∫f(x)dx is called the antiderivative of f(x). The integral of a constant c is cx, but otherwise ∫f(x)dx is not equal to f*x.