Why Does the Displacement Formula Include a 1/2 Factor in Physics?

[shinni]

I have just started out with Physics, so please don't expect to much :)

I've tried by myself to figure out where x = vt + ½t² comes from; I know (because I've read) how to deduce it from a v versus t graph, and that it only works with a constant acceleration.

My question is as follows:

If -for any instant in time- :
v = x / t
a = v / t
are true; Then, if i do this:

x = vt and v = at
Then subsitute the second in the first:
x = at²
I obviously miss the ½-factor from the area under the graph..
What does this formula (x=at²) tell me in this case? I don't really see why exactly this wouldn't also give me the displacement..

Could anyone be so kind to clear this up for me :) I'd hate to constantly have this question in the back of my mind.. just knowing that 'this is how it is'..

Thanks in advance!
Evert

 

[Doc Al]

If -for any instant in time- :
v = x / t

Since the speed is not constant, $\Delta x / \Delta t$ gives the average speed, not the final speed at time t.

For constant acceleration, the average speed is given by:
$$v_{ave} = (v_i + v_f)/2$$

So, if you start from rest (t=0, v_i=0, x=0), then $v_f = 2 v_{ave} = 2x/t$.

That should explain your missing ½-factor.

 

[m2003]

I appreciate your curiosity and desire to fully understand the concepts of 1D kinematics. The equation x = vt + ½t² comes from the definition of displacement, which is the change in position over time. In this case, we are considering motion with constant acceleration, so the equation simplifies to x = vt + ½at². This can also be written as x = (v + v0)t/2, where v0 is the initial velocity.

To understand why the ½ factor is necessary, let's look at the equation v = at. This equation tells us that the velocity at any given time is equal to the acceleration multiplied by the time. However, when we integrate this equation (which is essentially finding the area under the curve on a velocity vs. time graph), we get the equation x = ½at², which is the equation for displacement. The ½ factor is necessary because when we integrate, we are finding the average velocity over a given time interval, which is half of the final velocity.

So in summary, the equation x = at² tells us the displacement for an object with constant acceleration, but it is missing the ½ factor because it is not taking into account the average velocity over a given time interval. I hope this helps clear up any confusion and allows you to continue your journey in understanding physics. Keep asking questions and seeking knowledge!