# Relationship between initial velocity and final physics

### Physics Equations Page

Initial velocity of an object means the velocity at time t = 0. We know that velocity is the rate of change of displacement with time. Final velocity of a body is given. I thought I would chime in and give a slightly more fundamental approach in finding the solution. Starting with first principles, we will begin by taking the double. The initial velocity,vi is the velocity of the object before acceleration causes a change. After accelerating for some amount of time, the new velocity is the final.

When we start throwing things up into the air the convention is if something is moving up it is given a positive value, and if it is moving down we give it a negative value.

## Final and Initial Velocity

Well, for an object that is in free fall gravity would be accelerating it downwards, or the force of gravity is downwards. So, little g over here, if you want to give it its direction, is negative. Little g is So, we have the acceleration due to gravity.

The acceleration due to gravity ag is negative 9. Now I want to plot distance relative to time. Let's think about how we can set up a formula, derive a formula that, if we input time as a variable, we can get distance. We can assume these values right over here.

Well actually I want to plot displacement over time because that will be more interesting.

Right now we have something in terms of time, distance, and average velocity but not in terms of initial velocity and acceleration. We know that average velocity is the same thing as initial velocity vi plus final velocity vf over 2. We can only calculate Vavg this way assuming constant acceleration. Once again when were are dealing with objects not too far from the center of the earth we can make that assumption.

Assuming that we have a constant acceleration Once again we don't have what our final velocity is. So, we need to think about this a little more. We can express our final velocity in terms of our initial velocity and time. Just dealing with this part, the average velocity. So we can rewrite this expression as the initial velocity plus something over 2.

Well the final velocity is going to be your initial velocity plus your acceleration times change in time. Let me make sure that these are all vector quantities Hopefully it is ingrained in you that these are all vector quantities, direction matters.

And let's see how we can simplify this Well these two terms remember we are just dealing with the average velocity here These two terms if you combine them become 2 times initial velocity 2vi. All of this was another way to write average velocity. We still have to multiply this by this green change in time here.

All of this is what displacement is going to be. You have to parse the text of a problem for physical quantities and then assign meaning to mathematical symbols.

### Relation between initial velocity, final velocity and acceleration | Winner Science

The last part of this equation at is the change in the velocity from the initial value. Recall that a is the rate of change of velocity and that t is the time after some initial event. Rate times time is change. Move longer as in longer time. Acceleration compounds this simple situation since velocity is now also directly proportional to time.

Try saying this in words and it sounds ridiculous. Would that it were so simple. This example only works when initial velocity is zero.

Displacement is proportional to the square of time when acceleration is constant and initial velocity is zero. A true general statement would have to take into account any initial velocity and how the velocity was changing.

Physics - Mechanics: Projectile Motion (4 of 4)

This results in a terribly messy proportionality statement. Displacement is directly proportional to time and proportional to the square of time when acceleration is constant.

## Deriving displacement as a function of time, acceleration, and initial velocity

A function that is both linear and square is said to be quadratic, which allows us to compact the previous statement considerably. Displacement is a quadratic function of time when acceleration is constant Proportionality statements are useful, but not as concise as equations.

We still don't know what the constants of proportionality are for this problem. One way to figure them out is to use algebra. Start with the definition of velocity.