# How to interpret the equation of position with constant acceleration?

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x = x0 + v0 * t + 1/2 * constant acceleration * t^2

So this is supposed to be very very simple physics, but I still feel like there's a part of this equation I don't fully understand. The first term is the initial position of the body at t=0. The second term is the initial velocity at t=0, multiplied by the time elapsed since then. So far so good. Now, it's the third term I have trouble interpreting. I know it comes from integrating the acceleration function, but why exactly is it so?

I tried to reason it the following way: Let's assume a=2 m/s^2, and for the sake of simplicity make both the initial position and velocity zero. Ok, so if I want to know the position of this body at t=1, then given that constant acceleration, velocity at t=1 will be 2 m/s. So the body should have had a displacement of 2 metres, right?

But if I plug in the values in the formula, I'd get x = 1/2 * 2 m/s^2 * 1 s^2 = 1 metre. Why is this the case?

I know it must be pretty stupid, but I can't figure out what's wrong with my reasoning.
Thanks in advance!

## Answers and Replies

Doc Al
Mentor
Ok, so if I want to know the position of this body at t=1, then given that constant acceleration, velocity at t=1 will be 2 m/s.
Right.

So the body should have had a displacement of 2 metres, right?
Wrong. The velocity at t = 1 would be 2 m/s, but it starts out with v = 0. So the average speed over that first second is only 1 m/s.

Wrong. The velocity at t = 1 would be 2 m/s, but it starts out with v = 0. So the average speed over that first second is only 1 m/s.

I see. But what happens when t=2, for example? Because then the velocity at time 2 would be 4 m/s (the average over those two seconds will be 2 m/s), while the total displacement is 8 metres.

x = x0 + v0 * t + 1/2 * constant acceleration * t^2

So this is supposed to be very very simple physics, but I still feel like there's a part of this equation I don't fully understand. The first term is the initial position of the body at t=0. The second term is the initial velocity at t=0, multiplied by the time elapsed since then. So far so good. Now, it's the third term I have trouble interpreting. I know it comes from integrating the acceleration function, but why exactly is it so?

I tried to reason it the following way: Let's assume a=2 m/s^2, and for the sake of simplicity make both the initial position and velocity zero. Ok, so if I want to know the position of this body at t=1, then given that constant acceleration, velocity at t=1 will be 2 m/s. So the body should have had a displacement of 2 metres, right?

But if I plug in the values in the formula, I'd get x = 1/2 * 2 m/s^2 * 1 s^2 = 1 metre. Why is this the case?

I know it must be pretty stupid, but I can't figure out what's wrong with my reasoning.
Thanks in advance!
Acceleration is change of velocity with respect to time. In case of constant acceleration ##a##, it is simply ##a = \frac{ v_{final}-v_{initial}}{t_{final}-t_{initial}} \Rightarrow v_{final} =v_{initial} + a (t_{final}-t_{initial})##. If ##v_{initial} =0, t_{initial} = 0## and ##t_{final} = t##, then ##v_{final} = at##. But calculation displacement is a challenge here. Because here velocity is not constant. It varies with time (##v=at##). So you cannot write that displacement=##v_{final}t = at^2## or something like this. To counter this problem you have to assume that within every sufficiently small interval of time ##\Delta t##, the velocity remains constant. So within an interval ##\Delta t## displacement will be ##v\Delta t = at\Delta t##. Now if you add all the displacements at all intervals you will get total displacement which is ##\Sigma at \Delta t##. This is nothing but integration and the result is ##\frac{1}{2} at^2##.

rcgldr
Homework Helper
For constant acceleration, v = v0 + a t, and the average velocity at time t is ( (v0) + (v0 + a t) ) / 2 = v0 + 1/2 a t. Then the distance traveled is v0 t + 1/a 2 t^2 .

Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
x = x0 + v0 * t + 1/2 * constant acceleration * t^2

So this is supposed to be very very simple physics, but I still feel like there's a part of this equation I don't fully understand. The first term is the initial position of the body at t=0. The second term is the initial velocity at t=0, multiplied by the time elapsed since then. So far so good. Now, it's the third term I have trouble interpreting. I know it comes from integrating the acceleration function, but why exactly is it so?

I tried to reason it the following way: Let's assume a=2 m/s^2, and for the sake of simplicity make both the initial position and velocity zero. Ok, so if I want to know the position of this body at t=1, then given that constant acceleration, velocity at t=1 will be 2 m/s. So the body should have had a displacement of 2 metres, right?

But if I plug in the values in the formula, I'd get x = 1/2 * 2 m/s^2 * 1 s^2 = 1 metre. Why is this the case?

I know it must be pretty stupid, but I can't figure out what's wrong with my reasoning.
Thanks in advance!

If you plot a velocity vs. time graph you get a straight line with slope ##a##, the acceleration. The distance moved from times ##0## to ##t## is the area under the ##v## vs. ##t## graph between times ##0## and ##t## (do you see why?) So, the distance travelled between times ##0## and ##t## is the area of a triangle with base ##t## and height ##at##.

Matt2411
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
I see. But what happens when t=2, for example? Because then the velocity at time 2 would be 4 m/s (the average over those two seconds will be 2 m/s), while the total displacement is 8 metres.
I think you forgot the factor if ½ .

(½)(2)(22) = ½⋅2⋅4 = 4, not 8 .

Doc Al
Mentor
I see. But what happens when t=2, for example? Because then the velocity at time 2 would be 4 m/s (the average over those two seconds will be 2 m/s), while the total displacement is 8 metres.
At t = 2 the velocity will be 4 m/s. For those first 2 seconds the velocity went from zero to 4, for an average speed of 2 m/s. So the distance traveled would be 4 m, not 8.

SammyS
If you plot a velocity vs. time graph you get a straight line with slope ##a##, the acceleration. The distance moved from times ##0## to ##t## is the area under the ##v## vs. ##t## graph between times ##0## and ##t## (do you see why?) So, the distance travelled between times ##0## and ##t## is the area of a triangle with base ##t## and height ##at##.

I think this is what I was looking for. So the term ##\frac{1}{2} at^2## is really the area of a triangle given by the change in velocity with respect to the initial velocity.

Relating it to Doc Al's answer above (please correct me if I'm wrong), another way to look at it would be as the area of half the rectangle given by base ##t## and height ##at## (which would be the average velocity over that time interval, times the interval). Right?

I think you forgot the factor if ½ .

(½)(2)(22) = ½⋅2⋅4 = 4, not 8 .

At t = 2 the velocity will be 4 m/s. For those first 2 seconds the velocity went from zero to 4, for an average speed of 2 m/s. So the distance traveled would be 4 m, not 8.

Right, my bad. It's just I was trying to make sense of that 1/2 factor. Though I think I got it now with Ray's answer.

Thanks to all of you guys!

Doc Al
Mentor
Relating it to Doc Al's answer above (please correct me if I'm wrong), another way to look at it would be as the area of half the rectangle given by base ##t## and height ##at## (which would be the average velocity over that time interval, times the interval). Right?
Right. And that half rectangle is the triangle that Ray was talking about.

Matt2411
Right. And that half rectangle is the triangle that Ray was talking about.

Exactly. Thanks for the help, I really appreciate it

Since a picture is worth a thousand words.