Why displacement equals (average a) x (t)

[V0ODO0CH1LD]

I get the proof with the velocity as a function of time graphic.

I don't get why that is...

E.G.

Why does something with a constant acceleration of (1m/s)/s gets displaced the same as something with a constant velocity of .5m/s?

 

[rcgldr]

Why does something with a constant acceleration of (1m/s)/s gets displaced the same as something with a constant velocity of .5m/s?

That's only true for a specific amount of time, in this case 0 or 1 seconds, zero because no movement occurs, and 1 second because average velocity for the constant acceleration will be .5 m/s. At any other time, the average velocity and displacement will not be the same.

 

[V0ODO0CH1LD]

That's only true for a specific amount of time, in this case 0 or 1 seconds, zero because no movement occurs, and 1 second because average velocity for the constant acceleration will be .5 m/s. At any other time, the average velocity and displacement will not be the same.

Yeah, I realized that only after I posted...

But it still doens't answer the question of why that is..

Is it because the body spends the same amount of time accelerating towards the velocity of .5m/s as it spends accelerating away from it? And all of that at the same rate?

So it kind of compansates and adds up to the velocity which the body accelarates around?

 

[Ken G]

Yes. You are comparing vt with at²/2, so cancelling out one t means we are comparing v to at/2. Note that at is the final v, so if v starts out zero, then at/2 is the same thing as the average speed over the time interval t. So you will always have the same average v if you set t such that v = at/2.

 

[PJC01]

Displacement is a measure of how far an object has moved from its starting point. It is calculated by multiplying the average velocity of the object by the time it has been moving. This formula, displacement = (average velocity) x (time), applies to any type of motion, whether it is constant or changing.

In the case of an object with a constant acceleration, the average velocity can be calculated by taking the average of its initial velocity and final velocity. This average velocity is then multiplied by the time the object has been accelerating to determine the displacement.

For example, if an object starts with an initial velocity of 0 m/s and accelerates at a rate of 1 m/s^2 for 2 seconds, its final velocity would be 2 m/s. The average velocity would be (0 + 2)/2 = 1 m/s. Multiplying this average velocity by the time (2 seconds) gives a displacement of 2 meters.

Similarly, for an object with a constant velocity of 0.5 m/s, its average velocity would also be 0.5 m/s. Multiplying this by the time it has been moving would give the same displacement of 2 meters in 4 seconds.

Therefore, displacement is not dependent on the type of motion (constant acceleration or constant velocity), but rather on the average velocity and the time the object has been moving. This is why displacement equals (average velocity) x (time).