# Finding Position from Equations of Motion at Different Times

• PFuser1232

#### PFuser1232

Usually, equations of motion (with constant acceleration) are written in terms of values of position/velocity at time ##t=0##.
Take for example:

$$x = x_0 + v_0t + \frac{1}{2} a t^2$$

Where ##x_0## and ##v_0## are the values (at ##t=0##) of position and velocity respectively.

What if we're given position at some other time ##t=t_0##, instead of ##t=0##, and we're asked to find ##x## as a function of time?

What I do is I find a more general form of the equation I wrote above as follows:

$$\int_{x_0}^x dx' = \int_{t_0}^t v(t') dt'$$

Where ##x_0## now represents position at time ##t=t_0##.

Another way would be to eliminate the variable ##x(0)##

Is there any alternative approach?

Last edited:
This approach is great. Of course, what you would find is that you could also just substitute (t - t0) for t in the original equation (i.e., shift the time scale), assuming that v0 is the velocity at t = t0.

Chet

## 1. What is the equation for finding position from equations of motion at different times?

The equation for finding position from equations of motion at different times is x = x0 + v0t + 1/2at2, where x is the position, x0 is the initial position, v0 is the initial velocity, a is the acceleration, and t is the time.

## 2. How do I use this equation to find the position at a specific time?

To find the position at a specific time, plug in the values for x0, v0, a, and t into the equation x = x0 + v0t + 1/2at2. Make sure to use consistent units for all variables.

## 3. Can this equation be used for all types of motion?

Yes, this equation can be used for any type of motion as long as the acceleration is constant. This includes linear motion, circular motion, and projectile motion.

## 4. What is the significance of the initial position and velocity in this equation?

The initial position (x0) and velocity (v0) represent the starting point and speed of the object's motion. They are used to calculate the position at a specific time and provide a reference point for measuring the object's displacement.

## 5. How does the acceleration affect the position of an object over time?

The acceleration (a) determines how quickly the object's velocity changes over time, which in turn affects its position. If the acceleration is positive, the object will speed up and its position will increase. If the acceleration is negative, the object will slow down and its position will decrease.