What Is Initial Velocity?
Initial velocity — usually written u — is the speed and direction an object has at the start of the motion you're looking at, before any acceleration has had time to change it. Every kinematics problem has one: the speed of a ball the moment it leaves your hand, or the pace of a car just before the driver hits the brakes.
Why care about it? Because physics problems almost always describe motion by connecting three moments: where the object started, what happened in between, and where it ended up. Initial velocity pins down the starting moment, and once you know it, the whole motion becomes predictable.
The Initial Velocity Formula
The most basic equation of motion links starting and ending velocity:
[
v = u + at
]
Rearranging it to solve for the starting value gives:
[
u = v - at
]
Where:
- u is the initial velocity, in meters per second (m/s)
- v is the final velocity, in meters per second (m/s)
- a is the constant acceleration, in meters per second squared (m/s²)
- t is the time elapsed, in seconds (s)
The logic is easy to remember: whatever the object ended up doing, undo the acceleration to see where it began. Accelerating forward means it must have been going slower at the start; decelerating (negative a) means it was going faster at the start.
Worked Example: Braking Car
Say you clock a car at 15 m/s after it has spent 3 seconds accelerating gently at 2 m/s². How fast was it going before?
[
u = v - at = 15 \text{ m/s} - (2 \text{ m/s}^2 \times 3 \text{ s})
]
Breaking it down:
[
u = 15 \text{ m/s} - 6 \text{ m/s} = 9 \text{ m/s}
]
The car was traveling at 9 m/s when the acceleration began.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Initial Velocity | u | 9 | m/s |
| Final Velocity | v | 15 | m/s |
| Time | t | 3 | s |
| Acceleration | a | 2 | m/s² |
Try the calculator above with these numbers to confirm the result.
Interpreting Your Answer
The sign of u tells a story, not just a number:
- u > 0 — the object was already moving in the same direction it ends up going. Our braking-car example fits here.
- u = 0 — the object started from rest, like a dropped stone or a dragster at the starting line.
- u < 0 — the object was initially heading the opposite way. Throw a ball straight up while calling downward positive: the upward throw comes out negative, even though the ball obviously moves.
One caution: this formula only holds when acceleration is constant. Real-world cases with changing forces — rockets, air resistance, uneven braking — need different tools.
Quick Recap
- u = v − at comes straight from rearranging v = u + at.
- It works only for constant acceleration.
- A negative u means the object started moving the opposite way; zero means it started from rest.
- Keep units consistent: m/s, seconds and m/s² throughout.
If your problem gives distance instead of time, the final velocity calculator covers the companion equations of motion.
You might also like: Acceleration Calculator, Displacement Calculator, or Kinetic Energy Calculator.