# Homework Help: Second Order Differential Equation to show s = ut +(1/2)at^2

1. Dec 26, 2013

### Woolyabyss

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

If d^2s/dt^2 = a, given that ds/dt = u and s = 0, when t = 0, where a, u are constants

show that s = ut + .5at^2

2. The attempt at a solution

du/dt = a

cross multiplying and then integrating and we get

u = at

ds/dt = at

cross multiply and integrate

s = .5at^2

using limits when t = 0 then s = 0

I can't seem to get out the constant u

2. Dec 26, 2013

### Woolyabyss

could you maybe use v = u + at ? and say v = u + at but v = ds/dt

so ds/dt = u + at

cross multiplying and integrating and you get

s = ut + .5at^2

3. Dec 26, 2013

### CAF123

Yes, but first you need to derive this from the starting equality d2s/dt2 = a.

4. Dec 26, 2013

### Woolyabyss

I'm not sure how though could I say dv/dt = a

and then I'd get v = at but I just cant seem to get the u.

5. Dec 26, 2013

### rcgldr

After integration there's a constant. Say the initial velocity is u0, then

u = at + u0

The problem statement seems a bit off, if u = ds/dt, then there needs to be a constant for initial velocity in the equation such as u0:

s = u0 t + 1/2 a t^2

6. Dec 26, 2013

### Woolyabyss

But would using the limits not eliminate the constant of integration?

7. Dec 26, 2013

### rcgldr

The goal here is to produce a quadratic equation, not an intergral with limits.

8. Dec 26, 2013

### Woolyabyss

I got now thanks.