Question

How can you find the total displacement during the time interval of a graph?

please help totally lost :/

Answer 1

Area under v-t graph = total displacement

Step-by-step explanation:

The area under velocity time graph gives the total displacement of any object. Th attached figure shows a velocity time graph. In which the y axis shows the velocity and x axis shows the time taken.

The velocity is from 0 to 20 m/s and time is from 0 to 15 s. The total displacement is given by the area of region OABC.

So, d = 2 × area of triangle + area of rectangle

`d=2* (1)/(2)* 5* 20+5* 20`

d = 200 m

The total displacement covered by an object in this case is 200 m. Hence, the total displacement during the time interval of a graph can be determined by v-t graph.

Answer 2

Answer: in a space vs time graph, the displacement is the difference between the y coordinates of the initial and final points; in a velocity vs time graph, the displacement is given by the area included between the line describing the velocity, the x-axis and the vertical lines indicating the time interval of interest.

Space vs time graph
See the first picture attached.

Such a graph is a Cartesian plane with time on the x-axis and position on the y-axis describing the motion of an object.

The displacement between two moments in time t₁ and t₂ is calculated as the difference between the initial and the final position:
Δs = s(t₂) – s(t₁)

Therefore, the displacement is the difference between the y coordinates of the initial and final points.

Velocity vs time graph
This graph is very similar to the previous one, a Cartesian plane with time on the x-axis but velocity on the y-axis.

Let’s see the easiest case (see the second picture attached): the object moves at constant velocity, which will be graphed as a horizontal line.

In this kind of motion, we know that the displacement is given by the formula:
Δs = v × Δt

As you can see from the graph, this is the area of the rectangle that has as a base the time interval and as a height the velocity.

Similarly, if the object moves with accelerated motion with initial velocity not equal to zero (see the third picture attached), the displacement is given by the formula:
s = v₀·Δt + (1/2)·a·Δt²
= v₀·Δt + (1/2)·Δv ·t

which corresponds to the sum of the area of the triangle rectangle having as legs the time interval and the difference between the final and the initial velocity and the area of the rectangle having as a base the time interval and as a height the initial velocity.

We can then say that, in general, in a velocity vs time graph, the displacement is given by the area included between the line describing the velocity at any point, the x-axis and the vertical lines indicating the time interval of interest.