# Why displacement equals (average a) x (t)

• V0ODO0CH1LD
In summary, the question is about the displacement of an object with constant acceleration compared to an object with constant velocity, and the answer lies in the fact that the average velocity over a specific time interval will be the same for both, as long as the final velocity is half of the acceleration multiplied by the time interval. This is because at/2 is the same as the average speed over the time interval, and this will always be the same as long as the final velocity is half of the acceleration multiplied by the time interval.
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?

V0ODO0CH1LD said:
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.

rcgldr said:
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?

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.

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).

## 1. Why is displacement equal to average velocity multiplied by time?

The equation for displacement, d = v x t, is derived from the definition of velocity, which is the rate of change of displacement over time. In other words, velocity is the displacement divided by time. When we rearrange the equation to solve for displacement, we get d = v x t.

## 2. What is the significance of the average velocity in the displacement equation?

The average velocity in the displacement equation represents the constant rate at which an object is changing its position over a given time interval. It is the average of all the instantaneous velocities during that time period.

## 3. How is displacement different from distance travelled?

Displacement and distance travelled are two different measures of how far an object has moved. Displacement is a vector quantity that describes the straight-line distance and direction between an object's starting and ending positions. Distance travelled, on the other hand, is the total length of the path an object has taken, regardless of direction.

## 4. Does displacement always have to be in the same direction as the average velocity?

No, displacement and average velocity can be in different directions. For example, if an object moves forward for 5 meters and then moves backward for 5 meters, its displacement is 0 meters, but its average velocity is still non-zero because it has covered a distance of 10 meters in a certain amount of time.

## 5. Can displacement be negative?

Yes, displacement can be negative. This indicates that the object has moved in the opposite direction of the chosen reference point. For example, if an object moves -5 meters from its starting position, its displacement is -5 meters.

Replies
6
Views
858
Replies
13
Views
915
Replies
1
Views
649
Replies
23
Views
2K
Replies
8
Views
1K
Replies
2
Views
563
Replies
3
Views
719
Replies
2
Views
339
Replies
25
Views
2K
Replies
5
Views
2K