# Non-constant acceleration, solving for velocity

• turtles123
In summary, the velocity in terms of position is given by v=1/2*c*t^2+v0. To solve for the velocity in terms of only position, you will need to include a p0.
turtles123
Hello!

I am trying to solve for the velocity in terms of position of a particle moving with non-constant acceleration.
a=c*t (where c is a constant)

I can easily solve for velocity in terms of t.
dv/dt=a
dv/dt=c*t
I differentiate and get v=1/2*c*t^2+v0 (where v(0)=v0)

However I am not sure how to solve for velocity in terms of only position. I would know how to do this if acceleration was proportional to velocity, but since it is proportional to time, I am not sure what to do to get rid of the variable t.

Let me know if anyone has any suggestions.

What's position as a function of time (for this motion)?

Ibix
In general, knowing the acceleration function will only give you changes in velocity.
To know the velocity function, you will also need to know the initial velocity - or the velocity at some point in time.

We are not given the position as a function of time. We are only given acceleration and are to assume that v(0)=v0 (a constant) and x(0)=x0 (a different constant).

turtles123 said:
We are not given the position as a function of time.
If you are not given position as a function of time, you could always calculate it!

Dale
Yes, I did solve for position as a function of time, but then to make velocity of a function of position is very messy. When I plug in time= from the velocity equation into the position equation I get a very funky and long result, that I can't put in terms of x=

Alternatively if I try to solve for t from the x(t) equation it is very hard to do so because I have #t^3+#t.

turtles123 said:
Alternatively if I try to solve for t from the x(t) equation it is very hard to do so because I have #t^3+#t.
The general problem does look a bit gnarly. Why are you doing this?

turtles123 said:
Alternatively if I try to solve for t from the x(t) equation it is very hard to do so because I have #t^3+#t.
I thought you had a=ct and v0 = v0 ... c and V0 are constants.

If this is the case, you do not need to compute position as a function of time. If you did, you would need to include a p0.

dv(t)/dt = a(t)
So integrate.

But then I get velocity in terms of time and I need velocity in terms of position, x.

turtles123 said:
But then I get velocity in terms of time and I need velocity in terms of position, x.
So you will need a p0. So you solve p=f(t) - which will be a quadratic. The you solve for the quadratic.

1. You know that ##v=v_0+\frac{1}{2}ct^2.##
2. Solve this equation for ##t## in terms of ##v## and the constants.
3. Observe that $$a=\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}=v\frac{dv}{dx}~\Rightarrow~\frac{dv}{dx}=\frac{a}{v}=\frac{ct}{v}.$$4. 4. Replace ##t## with what you got in step 2.
5. Separate variables and integrate.

## 1. What is non-constant acceleration?

Non-constant acceleration refers to the change in velocity of an object that is not constant. This means that the object is either speeding up or slowing down at a varying rate.

## 2. How do you calculate velocity with non-constant acceleration?

In order to calculate velocity with non-constant acceleration, you will need to use the formula v = u + at, where v represents final velocity, u represents initial velocity, a represents acceleration, and t represents time.

## 3. Can you solve for velocity without knowing acceleration?

No, in order to solve for velocity, you will need to know the acceleration of the object. Without this information, you will not be able to accurately calculate the velocity.

## 4. How does non-constant acceleration affect an object's motion?

Non-constant acceleration can greatly affect an object's motion. If the acceleration is positive, the object will be speeding up. If the acceleration is negative, the object will be slowing down. This can also cause the object's motion to be curved or uneven.

## 5. What are some real-life examples of non-constant acceleration?

Some real-life examples of non-constant acceleration include a car speeding up or slowing down, a rollercoaster going up and down hills, and a ball being thrown into the air and falling back down. Essentially, any situation where an object's velocity is changing at a varying rate is an example of non-constant acceleration.

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