# How To Find The Final Velocity Of Any Object

Many formulae and equations in physics problems involve calculating an initial and final velocity. The difference between initial and final velocity in equations for conservation of momentum or equations of motion tell you the velocity of an object before and after something happens. This could be a force applied to the object, a collision or anything that could change its trajectory and motion.

The final velocity of an object is an instantaneous velocity at a certain time t at the end of an interval. It measures the final speed (with a directional component) after a given time.

To calculate final velocity for an object under uniform acceleration, you can use the corresponding equation of motion. These equations use combinations of distance, initial velocity, final velocity, acceleration and time to relate them to one another.

## Final Velocity Formula

Determining changes in velocity relies on the acceleration of an object. Acceleration is the change in velocity over time. For example, the final velocity (v_{f} ) formula that uses initial velocity (_v_{i}*), acceleration (*a*) and time (*t_) is:

\(v_f = v_i + a\Delta t\)

For a given initial velocity of an object, you can multiply the acceleration due to a force by the time the force is applied and add it to the initial velocity to get the final velocity. The "delta" Δ in front of the *t* means it's a change in time that can be written as *t*_{f} *− t*_{i}.

### TL;DR (Too Long; Didn't Read)

This formula only works with a constant acceleration (or in the case of variable acceleration, an average acceleration over the time interval).

This is ideal for a ball falling toward the ground due to gravity. In this example, the acceleration due to the force of gravity would be the gravitational acceleration constant *g* = 9.8 m/s^{2}. This acceleration constant tells you how fast any object accelerates when you drop it on Earth, no matter what the mass of the object is.

If you drop a ball from a given height and calculate how long it takes the ball to reach the ground, then you can determine the velocity just before it hits the ground as the final velocity. The initial velocity would be 0 if you dropped the ball without any external force. Using the equation above, you can determine the final velocity of the object _v_{f}_.

## Alternative Final Velocity Calculator Equations

You can always use the other kinematic equations for whichever situation you're working with, as they are always logically and mathematically equivalent with one another. If you knew the distance an object traveled (Δ*x*), along with the initial velocity and time it took to travel that distance, you could calculate final velocity using the equation for final position after an elapsed time, based on the average velocity:

\(\Delta x = \bar{v} t = \frac{1}{2}(v_f+v_i)t

\ \newline{} \

\text{This can then be rearranged to solve for } v_f:

\ \newline{} \

v_f = \frac{2\Delta x}{t} – v_i\)

Make sure to use the correct units in these calculations.

## A Rolling Cylinder

For a cylinder rolling down an inclined plane or a hill, you can calculate the final velocity using the formula for conservation of energy. This formula dictates that, if the cylinder starts from rest, the energy it has at its initial position should equal its energy after rolling down a certain distance.

At its initial position, the cylinder has no kinetic energy because it's not moving. Instead, all of its energy is potential energy, meaning its energy can be written as:

\(E = mgh\)

with a mass *m*, gravitational constant *g* = 9.8 m/s^{2} and height *h*. After the cylinder has rolled down a distance to a height of *h* = 0, its energy is only the sum of its translational kinetic energy and rotational kinetic energy. This gives you:

\(E = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2\)

for velocity *v*, rotational inertia *I* and angular velocity "omega" *ω*.

The rotational inertia *I* for a cylinder is *I* = *mr*^{2} / 2 and the angular velocity *ω* = v/r. By the law of conservation of energy, you can set the cylinder's initial potential energy equal to the sum of the two kinetic energies. Solving for *v*, we can find:

\(mgh = \frac{1}{2}mv^2 + \frac{1}{2}(\frac{mr^2}{2})(\frac{v^2}{r^2})

\ \newline{} \

\text{the mass and radius cancel leaving:}

\ \newline{} \

gh = \frac{1}{2}v^2 + \frac{1}{4}v^2 = \frac{3}{4}v^2

\ \newline{} \

\text{solving for $v$, we get:}

\ \newline{} \

v = \sqrt{\frac{4}{3}gh}\)

This formula for the final velocity doesn't depend on the weight or mass of the cylinder. If you knew the weight of the cylinder formula in kg (technically, the mass) for different cylindrical objects, you could compare different masses and find their final velocities are the same, because mass cancels out of the expression above.

## Other Influences on Velocity

Anything that affects the acceleration of a moving object will affect the state of velocity (or constant acceleration). If a force is applied in the opposite direction of the velocity, then the acceleration is negative relative to the velocity, so the final velocity will be lower than the initial velocity.

A more complex scenario might be found with an object in free fall. If an object is falling under the force of gravity it will also experience the force of air resistance in the opposite direction. Eventually, in a complex relationship of velocities and forces, the object will actually reach a constant velocity known as the terminal velocity for a falling object.

### References

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Ather, S. Hussain. "How To Find The Final Velocity Of Any Object" *sciencing.com*, https://www.sciencing.com/final-velocity-object-5495923/. 29 March 2023.

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