Displacement Calculator

Enter an initial velocity, acceleration, and elapsed time to find net displacement (s = ut + ½at²), final velocity (v = u + at), average velocity, and total distance traveled.

Quick Facts

Displacement formula
s = ut + ½at²
Net, signed change in position — a vector, not a distance.
Final velocity
v = u + at
Velocity at the end of the time interval under constant acceleration.
Displacement vs. distance
Distance ≥ |Displacement|
They match only if the object never reverses direction during the interval.
Time-free form
v² = u² + 2as
Useful when elapsed time is unknown but both velocities and acceleration are known.

Your Results

Calculated
Net Displacement
-
s = u·t + ½·a·t² (signed)
Final Velocity
-
v = u + a·t
Average Velocity
-
(u + v) / 2 = s / t
Total Distance Traveled
-
Path length; exceeds |displacement| if direction reverses

Ready

Enter initial velocity, unit system, acceleration, and time elapsed, then press Calculate.

How to Calculate Displacement in Constant-Acceleration Motion

Displacement is the straight-line, signed change in an object's position — how far it ended up from where it started, and in which direction, regardless of the path taken to get there. For motion under constant acceleration (free fall, braking, a car accelerating from a stoplight), displacement follows directly from three quantities: initial velocity, acceleration, and elapsed time. This calculator applies the standard kinematics equation s = ut + ½at² along with the companion equations v = u + at (final velocity) and the average-velocity relationship v̄ = (u + v) / 2, and also reports the total distance traveled, which can differ from displacement if the object changes direction during the interval.

Where s = ut + ½at² comes from

Under constant acceleration, velocity changes linearly with time: v(t) = u + at. Displacement equals the area under the velocity-vs-time graph, which for a straight line is the area of a trapezoid with parallel sides u and v over base t: s = ½(u + v)·t. Substituting v = u + at gives s = ½(u + u + at)·t = ut + ½at² — the form this calculator uses. A third, time-free form follows from eliminating t between these two equations: v² = u² + 2as, useful whenever the elapsed time isn't known directly.

Displacement vs. distance traveled

Displacement is a vector — it can be positive, negative, or zero, and depends only on the start and end positions. Distance traveled is a scalar — the total length of the path covered — and it can never decrease. The two match only when the object moves in a single direction throughout the interval. If acceleration opposes the initial velocity strongly enough that the object slows, stops, and reverses (a ball thrown upward that falls back down, or a car braking hard enough to roll backward), this calculator splits the motion at the instant velocity crosses zero and adds the two path segments together to report the correct total distance, which will be larger than the magnitude of the net displacement.

Sign conventions and practical notes

  • Choose a positive direction before you start (commonly "forward" or "up") and keep initial velocity, acceleration, and the resulting displacement consistent with that choice — a negative acceleration simply points in the negative direction, it doesn't necessarily mean the object is slowing down.
  • Free fall near Earth's surface uses a ≈ −9.8 m/s² (or −32.2 ft/s²) when "up" is chosen as positive.
  • These equations assume constant acceleration; they do not apply once acceleration varies with time (air resistance at high speed, jerk-limited motion, or other non-uniform forces).
  • If you only know average velocity and time — not acceleration — use the simpler relationship s = v̄ × t instead.

Frequently Asked Questions

What is the formula for displacement in physics?
For motion with constant acceleration, displacement is s = ut + ½at², where u is initial velocity, a is acceleration, and t is elapsed time. This gives the net, signed change in position — not the total distance traveled.
What is the difference between displacement and distance traveled?
Displacement is a vector: the straight-line change in position from start to end, which can be positive, negative, or zero. Distance is a scalar: the total length of the path traveled, which is always positive. The two are equal only if the object moves in one direction the whole time; if it reverses direction, distance traveled is greater than the magnitude of displacement.
How do I find displacement without knowing the time?
Use v² = u² + 2as, rearranged to s = (v² − u²) / (2a), where v is the final velocity. This form is useful whenever you know initial velocity, final velocity, and acceleration but not how long the motion took.
Can displacement be negative?
Yes. Displacement is measured relative to a chosen positive direction. If an object ends up behind its starting point, its displacement is negative, even though the distance it traveled is always reported as a positive number.

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