What is the final velocity?

How it works

Five variables: initial velocity v₀, final velocity v, acceleration a, time t and displacement s. Type any three; the page finds the other two from the equations of motion for constant acceleration (OpenStax Equations 3.10 to 3.14):

1. v = v₀ + a t
2. s = ½ (v₀ + v) t
3. s = v₀ t + ½ a t²
4. s = v t − ½ a t²
5. v² = v₀² + 2 a s

Average velocity = s ÷ t = (v₀ + v) ÷ 2.

Two answers. Equation 5 gives v (or v₀) as ±√(…), and equations 3 and 4 give t as the roots of a quadratic, t = (−v₀ ± √(v₀² + 2as)) ÷ a or t = (v ± √(v² − 2as)) ÷ a. Every root that fits the bounds below is shown. With a = 0, t = s ÷ v₀.

Rules. Time is more than 0 and at most 1,000 s. Velocities are from −100,000 to 100,000 m/s, acceleration from −100 to 100 m/s² (about ±10 g), and displacement from −10,000,000 to 10,000,000 m. A typed value outside its range gets a message on its box; a solved value outside its range is no answer. If no value in range fits (for example v² = v₀² + 2as below zero, a time that would be negative, or a = 0 when the time is asked from v₀ and v), the page shows no answer and names the equation. Typed values that do not agree with each other (all of v₀, v and t plus an a that does not fit) give an "inconsistent" message.

Output format. Decimals and significant figures below are the most shown; trailing zeros are dropped (23.0 shows as 23, money keeps its cents). Velocities, typed or solved, and the average velocity show 6 significant figures in the unit you pick (m/s by default). A solved time or displacement shows 4 significant figures, but whole units are never rounded away (1,234.5 m shows 1,235 m); a solved acceleration shows at most 6 decimals. The values are computed in double precision in SI units (m, s, m/s); units convert exactly (1 ft = 0.3048 m, 1 mi = 1,609.344 m, 1 kn = 1,852 m per hour).

Assumptions

• The acceleration is constant over the whole time.
• The motion is along one straight line; a negative velocity, acceleration or displacement points the other way.

Worked examples by hand

A landing jet (OpenStax Example 3.7). v₀ = 70 m/s, a = −1.5 m/s², t = 40 s: v = 70 − 60 = 10 m/s; s = ½ × (70 + 10) × 40 = 1,600 m; average 40 m/s.

A dragster (Example 3.8). v₀ = 0, a = 26 m/s², t = 5.56 s: s = ½ × 26 × 5.56² = 13 × 30.9136 = 401.8768 m; v = 26 × 5.56 = 144.56 m/s.

The dragster by distance (Example 3.9). v₀ = 0, a = 26 m/s², s = 402 m: v = √(2 × 26 × 402) = √20,904 = 144.582 m/s; t = v ÷ a = 5.56085 s.

Braking over a distance. v₀ = 20 m/s, v = 5 m/s, s = 50 m: a = (25 − 400) ÷ 100 = −3.75 m/s²; t = 2 × 50 ÷ 25 = 4 s; average 12.5 m/s.