All solvers

    How to use v² − u² = 2as

    By Engr. Mehmood Kakar · Updated

    Quick answer

    Use v² − u² = 2as when acceleration is constant and the question has no time in it. Write down what you know (u, v, a, s), rearrange for the one you need, square the velocities before adding or subtracting, and take a square root at the end if you are finding a velocity. Slowing down means a is negative.

    When to use it

    v² − u² = 2as works for an object with uniform acceleration, and it is the equation to reach for when a question gives you velocities, acceleration and distance but says nothing about time. It is on the equation sheet, so the skill being tested is choosing it and rearranging it.

    Step 1: list u, v, a and s

    Before any maths, write the four letters down the side of your page and fill in what the question gives you. Watch for words that hide a value:

    • "starts from rest" or "is dropped": u = 0
    • "comes to rest" or "stops": v = 0
    • "falls freely": a = 9.8 m/s²
    • "decelerates at 4 m/s²": a = −4 m/s²

    The one letter left blank is what you need to find.

    Finding final velocity

    A car starts from rest and accelerates at 3.0 m/s² over 24 m. Calculate its final velocity. (3 marks)

    1. u = 0, a = 3.0 m/s², s = 24 m, v = ?
    2. v² = u² + 2as = 0² + 2 × 3.0 × 24 = 144
    3. v = √144
    4. v = 12 m/s

    Stopping at 144 is the most common lost mark. That is v², not v.

    Finding distance

    A cyclist speeds up from 8 m/s to 20 m/s with an acceleration of 2 m/s². How far does she travel? (3 marks)

    1. u = 8 m/s, v = 20 m/s, a = 2 m/s², s = ?
    2. s = (v² − u²) ÷ 2a = (20² − 8²) ÷ (2 × 2)
    3. s = (400 − 64) ÷ 4 = 336 ÷ 4
    4. s = 84 m

    Square each velocity separately. (20 − 8)² = 144 is not the same as 20² − 8² = 336.

    Braking and deceleration

    A car travelling at 72 km/h brakes with a deceleration of 4 m/s². Calculate its braking distance. (4 marks)

    1. Convert: u = 72 ÷ 3.6 = 20 m/s; v = 0; a = −4 m/s²
    2. s = (v² − u²) ÷ 2a = (0² − 20²) ÷ (2 × −4)
    3. s = −400 ÷ −8
    4. s = 50 m

    The two minus signs cancel, giving a positive distance. If you get a negative distance, the sign of a is wrong. Braking distance goes up with the square of the speed: double the speed and the braking distance is four times as long.

    Falling objects

    A stone is dropped from a height of 45 m. Ignoring air resistance, calculate its speed just before it hits the ground. a = 9.8 m/s². (3 marks)

    1. u = 0, a = 9.8 m/s², s = 45 m, v = ?
    2. v² = 0² + 2 × 9.8 × 45 = 882
    3. v = √882 = 29.698…
    4. v = 29.7 m/s

    Finding initial velocity

    A train reaches 30 m/s after accelerating at 2 m/s² over 200 m. What was its initial velocity? (3 marks)

    1. v = 30 m/s, a = 2 m/s², s = 200 m, u = ?
    2. u² = v² − 2as = 30² − 2 × 2 × 200 = 900 − 800 = 100
    3. u = √100
    4. u = 10 m/s

    Check any answer with the v² − u² = 2as calculator. It shows every squaring and square-root step.

    Practice

    1. A motorbike accelerates from rest at 4.0 m/s² over 50 m. Calculate its final velocity.

    v² = 0 + 2 × 4.0 × 50 = 400 → v = √400 = 20 m/s

    2. A car slows from 15 m/s to rest over 45 m. Calculate its acceleration.

    a = (0² − 15²) ÷ (2 × 45) = −225 ÷ 90 = −2.5 m/s² (a deceleration of 2.5 m/s²)

    3. A car at 30 m/s brakes with a deceleration of 6 m/s². Calculate its braking distance.

    s = (0² − 30²) ÷ (2 × −6) = −900 ÷ −12 = 75 m

    Frequently asked questions

    What does each letter in v² − u² = 2as stand for?

    v is the final velocity and u the initial velocity (both in m/s), a is the acceleration in m/s², and s is the distance in metres.

    How do I know when to use v² − u² = 2as?

    Use it when the acceleration is uniform and the question involves velocities, acceleration and distance but not time. If time is involved, a = Δv/t is usually the one you need.

    Why is my acceleration negative?

    A negative acceleration means the object is slowing down. That is correct for braking questions: the size of the deceleration is the number without the minus sign.

    What does "starts from rest" mean?

    The initial velocity is zero, so u = 0. "Comes to rest" or "stops" means the final velocity is zero, so v = 0.

    Engr. Mehmood Kakar

    Written by Engr. Mehmood Kakar

    Physics educator and engineer (MSc Physics, B.Ed, M.Ed) with 8 years of hands-on teaching, from college students to teacher training. About the author · LinkedIn (opens in a new tab)

    Checked against the AQA GCSE Physics (8463) specification on 14 November 2026.