# Find the Velocity when accelration is not constant

1. Jan 26, 2013

### Northbysouth

Find the Velocity when acceleration is not constant

1. The problem statement, all variables and given/known data

A particle moves along the positive x-axis with an acceleration in meters per second squared which increases linearly with x expressed in millimeters, as shown on the graph for an interval of its motion. If the velocity of the particle at x = 40 mm is 0.36 m/s, determine the velocity at x = 130 mm.

I have attached an image of the question.

2. Relevant equations

3. The attempt at a solution
vx= v0x+at
or
s = s0 + v0xt +1/2*at2

But due to the acceleration not being constant these equations are not suitable. I'm guessing that I need an integral but I'm not sure how to start. Any help would be appreciated.

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Last edited: Jan 26, 2013
2. Jan 26, 2013

### Staff: Mentor

Consider the work done by the force that is accelerating the particle.

3. Jan 26, 2013

### rude man

Well, you have a(x) = bx with b given.
You also have a = dv/dt = dv/dx * dx/dt = v dv/dx
So v dv/dx = bx.

This diff. eq. can easily be solved by separation of variables. Remember the constant of integration which you determine from the given initial condition ...

4. Jan 26, 2013

### voko

You surely meant a(x) = bx + c.

5. Jan 26, 2013

### rude man

I surely did.

Same approach, though.

6. Jan 26, 2013

### Northbysouth

I've only recently started learning about differential equations so bear with me.

From what I understand I should get:

v dv/dx = bx + C

v dv = (bx + C)dx

∫v dv = ∫(bx + C)dx

v2/2 = bx2/2 + Cx + D

I then know that v(0.040 meters) = 0.36 m/s and that I can use this information to find D but how do I find b and D? Is b just the gradient of the slope?

7. Jan 26, 2013

### voko

Pick any two values of x. Then, since a = bx + c, you should get two linear equations for b and c with known values of a and x. Solve them.

8. Jan 26, 2013

### rude man

Yes, voko has shown you how you can get b and c. Remember y = (slope)*x + y-intercept from high school algebra? In this case b is the slope and c is the a-intercept.

You did well with solving the diff. eq. BTW.

9. Jan 26, 2013

### Northbysouth

I got it. The answer is 1.0366 m/s.

Thank you to everyone for your help.

10. Jan 27, 2013

### Staff: Mentor

Just for fun, here's how I would have solved it using the method I hinted at in my earlier post:

ΔKE = ∫F.dx
Δ(v2/2) = ∫a.dx

The integral is the area under the curve, so you can just read it off of your diagram. No need for any formal calculus.

The methods are entirely equivalent, of course. And until you are comfortable with the math, probably best to work things out as you did.