# Finding equation of the position

• Lord Dark
In summary, the conversation discusses a velocity versus time graph and how to use it to find the position, displacement, and distance of a particle at different times. The equation for position as a function of time is given, and the concept of integration is introduced to find the position from the velocity graph. The conversation also discusses the role of a constant of integration and how visual inspection can also be used to solve the problem.
Lord Dark

## Homework Statement

hi everyone ,, this is my last question before exam :

1) The figure shows the velocity versus time graph. If a particle starts motion at t=0.0
from x0 = 10.0 m:
a) Write the equation describing the position of the particle as a function of time t.
b) What are the positions of the particle at t= 2 s and t = 4 s.?
c) What is the position of the particle at the instant when it momentarily stops?
d) Find the distance and displacement of the particle between t = 2 s and t = 5 s.

## The Attempt at a Solution

I know that a=0 ,, and at t=0 > x=10m and at the same time Vo=12 m/s ,, and the velocity is decreasing so I made this equation :x=10-12t but the problem at t=0 Vo=0 and when I apply x(3) I don't get 18m which it's what i have to get by the graph .. can someone tell me what to do to get the equation ..?

#### Attachments

• Q1.JPG
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Consider how you find displacement from a velocity time graph.

If V = dX/dt, then isn't

X = ∫ V dt ?

And if you integrate V on your graph, what is the easiest way to do that?

Can you think of something that rhymes with area?

LowlyPion said:
Consider how you find displacement from a velocity time graph.

If V = dX/dt, then isn't

X = ∫ V dt ?

And if you integrate V on your graph, what is the easiest way to do that?

Can you think of something that rhymes with area?
I didn't understand you quite well ,, I tried by doing X = ∫ V dt from x(0,3) x=18 and i get v=6 m/s ,, and from x(0,5) v=2 m/s but the problem is at x(0,0) I should get x=10 and I don't get it,, V changes ...

Lord Dark said:
I didn't understand you quite well ,, I tried by doing X = ∫ V dt from x(0,3) x=18 and i get v=6 m/s ,, and from x(0,5) v=2 m/s but the problem is at x(0,0) I should get x=10 and I don't get it,, V changes ...

Xo is the constant of integration isn't it?

yea Xo is constant ,, should I include it in the integral or X = Xo + ∫ V dt ??
and BTW when i integrate ∫ V dt ,, I'll get Vt right ??

Lord Dark said:
yea Xo is constant ,, should I include it in the integral or X = Xo + ∫ V dt ??

V(t) = 12 - 4t isn't it?

So ∫ V(t) dt = ∫ (12 - 4t) dt = X(t)

That means that X(t) = 12t - 1/2*4*t2 + C

Where C = Xo

aha ,, so that what you meant X = ∫ V ,, I didn't know that first I should get V(t) equation then integrate .. thanks very much :D

Lord Dark said:
aha ,, so that what you meant X = ∫ V ,, I didn't know that first I should get V(t) equation then integrate .. thanks very much :D

You can also solve through visual inspection, by simply calculating the area under the curve. Each square represents 4 m.

## 1. How do you find the equation of an object's position?

To find the equation of an object's position, you need to know its initial position, velocity, and acceleration. You can then use the equation of motion, x = x0 + v0t + 1/2at^2, to calculate the position at any given time.

## 2. What is the importance of finding the equation of an object's position?

Knowing the equation of an object's position allows you to accurately predict its future position and track its motion over time. It is also essential for analyzing and understanding the behavior of objects in motion.

## 3. Can the equation of position be used for any type of motion?

Yes, the equation of position can be used for any type of motion that follows a constant acceleration, such as freefall or projectile motion. It is also applicable to circular motion if the object's velocity and acceleration are constant at any given point.

## 4. How do you incorporate direction into the equation of position?

Direction can be incorporated into the equation of position by using vector notation. The position, velocity, and acceleration vectors will have both a magnitude and a direction, which can be represented using appropriate symbols or unit vectors.

## 5. Can the equation of position be used for objects with changing acceleration?

Yes, the equation of position can be modified to account for changing acceleration. In this case, the equation would be x = x0 + v0t + 1/2∫(a(t)dt), where a(t) is the function for acceleration at any given time. This equation is more commonly used in advanced physics and calculus courses.

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