How to Find the Final Velocity of any Object

Many formulae and equations in physics 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.

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

For example, the final velocity (vf ) formula that uses initial velocity (vi), acceleration (a) and time (t) is:

v_f = v_i + aΔt.

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 tf − ti.

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/s2. This acceleration constant tells you how fast any object accelerates when you drop it on Earth, no matter the 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 final velocity vf.

Alternative Final Velocity Calculator Equations

You can use the other kinematic equations as appropriate for whichever situation you're working with. 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:

v_f = \frac{2Δx}{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/s2 and height h. After the cylinder has rolled down a distance, its energy is the sum of its translational kinetic energy and rotational kinetic energy. This gives you:

E = \frac{1}{2} mv^2 + \frac{1}{2}Iω^2

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

The rotational inertia I for a cylinder is I = mr2/ 2. 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, you obtain

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.

References

About the Author

S. Hussain Ather is a Master's student in Science Communications the University of California, Santa Cruz. After studying physics and philosophy as an undergraduate at Indiana University-Bloomington, he worked as a scientist at the National Institutes of Health for two years. He primarily performs research in and write about neuroscience and philosophy, however, his interests span ethics, policy, and other areas relevant to science.