# Difficulty understanding formula and area under v-t graph

• aboojoo
In summary, displacement is average velocity times change in time, where average velocity is the sum of the initial and final velocities all divided by 2.
aboojoo
1. The problem

For sake of format I attached the a screenshot of the course material I'm having difficulty wrapping my walnut around. Which is how:

Total Displacement = Area of Triangle + Area of Rectangle
or
Δvector d = Atriangle + ARectangle
or
Δvector d = 1/2 (V2-V1)Δt +V1*Δt

Translates in to:

Δvector d= 1/2 (V1+V2)Δt

Could someone explain it to me? I feel as if I'm blatantly missing an obvious answer as to why this is but since there is no answer in the text, I'm reaching out.

help.png contains screenshot
q.doc contains my question in better formatting

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Is it the algebra you are having trouble with? Or the concept?
1/2k( a -b) + kb =
1/2ka - 1/2kb + kb =
1/2ka. + 1/2kb =
1/2k(a + b)

Conceptually, for constant acceleration , displacement is average velocity times change in time, where average velocity is the sum of the initial and final velocities all divided by 2.

1 person
Hi aboojoo! http://img96.imageshack.us/img96/5725/red5e5etimes5e5e45e5e25.gif

If velocity were constant, then the graph would be a horizontal line, and you'd be happy to multiply that velociy value by Δt to find the displacement during that interval of time, Δt. You can see that by performing that multiplication you are actually finding the area of a rectangle, the area under that graph. The same property holds even where the velocity is not constant; you could think of dividing it up into lots of thin vertical rectangles if you wish, where velocity is almost constant during each narrow time interval. However you look at it, you are still determinng the area under the curve during that whole time interval. Once you appreciate this, you can use whatever method is easiest to determine the area under the graph, e.g., a large rectangle with a triangular piece atop it, as illustrated in this problem.

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1 person
I totally understand now , thanks a lot guys you've really helped, both conceptually and algebraic.

I can understand how this concept can be confusing at first. Let me try to break down the formula and the concept of area under a v-t graph for you.

Firstly, the v-t graph represents the velocity (v) of an object over time (t). The area under the graph represents the displacement (d) of the object during that time period. This is because displacement is the distance an object has traveled in a specific direction.

Now, let's look at the formula. The total displacement (Δvector d) is equal to the area of a triangle (Atriangle) plus the area of a rectangle (ARectangle). This is because the graph can be divided into two parts - the triangle and the rectangle.

The triangle represents the change in velocity (V2-V1) over a specific time interval (Δt). This is why the formula includes 1/2, as it is half of the base (change in time) multiplied by the height (change in velocity).

The rectangle represents the initial velocity (V1) multiplied by the time interval (Δt). This is because the initial velocity stays constant throughout the time interval.

Therefore, the final formula is Δvector d = 1/2 (V2-V1)Δt + V1*Δt, which can also be written as Δvector d = 1/2 (V1+V2)Δt.

I hope this explanation helps you understand the formula and the concept better. It's important to remember that in science, formulas are just tools to help us understand and quantify natural phenomena. Keep practicing and applying these concepts, and you will become more comfortable with them.

## 1. Why is it important to understand the formula for area under a v-t graph?

Understanding the formula for area under a v-t graph is important because it allows us to calculate the distance traveled by an object over a specific time period. This is crucial in many scientific fields, such as physics and engineering.

## 2. What is the formula for calculating area under a v-t graph?

The formula for calculating area under a v-t graph is A = ∫v(t) dt, where A is the area, v(t) is the velocity at a given time, and dt is the change in time.

## 3. How do I interpret the area under a v-t graph?

The area under a v-t graph represents the displacement or distance traveled by an object. The sign of the area (positive or negative) indicates the direction of the object's motion.

## 4. How can I improve my understanding of the formula and area under a v-t graph?

To improve your understanding of the formula and area under a v-t graph, it is important to practice solving problems and visualizing the concept. You can also seek help from a tutor or teacher if you are struggling to understand.

## 5. What are some real-life applications of understanding area under a v-t graph?

Understanding area under a v-t graph has many real-life applications, such as calculating the distance traveled by a car over a certain time period, determining the displacement of an object in projectile motion, and analyzing the speed and acceleration of an athlete in a race.

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