# The kinematics equations

Canonical: https://duckyhelper.com/learn/physics/kinematics-equations/
Updated: 2026-10-01

The kinematics equations describe motion with constant acceleration. They connect five quantities: displacement \(\Delta x\), initial velocity \(v_0\), final velocity \(v\), acceleration \(a\) and time \(t\). Each equation leaves out one of the five. Write down the three you know, pick the equation that leaves out the one you neither know nor need, then solve. Choose a positive direction and stick to it.

## Key ideas

$$
v = v_0 + at
$$

$$
\Delta x = v_0 t + \tfrac{1}{2} a t^2
$$

$$
v^2 = v_0^2 + 2a\,\Delta x
$$

$$
\Delta x = \tfrac{1}{2}(v_0 + v)\,t
$$

- \(\Delta x\) is displacement, the change in position, in meters (m). It can be negative.
- \(v_0\) is the initial velocity and \(v\) is the final velocity, in meters per second (m/s).
- \(a\) is the acceleration, in meters per second squared (m/s²). It must be constant for these equations to work.
- \(t\) is the time, in seconds (s).

**Pick the equation by the quantity you don't have**

| Equation | Leaves out | Use it when |
| --- | --- | --- |
| \(v = v_0 + at\) | \(\Delta x\) | No distance is given or asked |
| \(\Delta x = v_0 t + \tfrac{1}{2}at^2\) | \(v\) | No final velocity is given or asked |
| \(v^2 = v_0^2 + 2a\Delta x\) | \(t\) | No time is given or asked |
| \(\Delta x = \tfrac{1}{2}(v_0 + v)t\) | \(a\) | No acceleration is given or asked |

**Free fall** is the same math in the vertical direction. If you take up as positive, \(a = -g = -9.8\ \text{m/s}^2\) the whole time an object is in the air, on the way up, at the top and on the way down.

> **Tip: Hidden values**
>
> "Starts from rest" means \(v_0 = 0\). "Comes to a stop" means \(v = 0\). "Dropped" means \(v_0 = 0\). At the top of a throw straight up, \(v = 0\).

## Worked examples

**Example 1: speeding up from rest**

Problem: A car starts from rest and speeds up at 3.0 m/s² for 5.0 s. How fast is it going, and how far does it travel?

1. Knowns: \(v_0 = 0\), \(a = 3.0\ \text{m/s}^2\), \(t = 5.0\ \text{s}\).
2. Final speed, using the equation without \(\Delta x\).

   $$
   v = 0 + (3.0\ \text{m/s}^2)(5.0\ \text{s}) = 15\ \text{m/s}
   $$
3. Distance, using the equation without \(v\).

   $$
   \Delta x = 0 + \tfrac{1}{2}(3.0\ \text{m/s}^2)(5.0\ \text{s})^2 = 37.5\ \text{m}
   $$
4. The data have 2 significant figures, so round to 38 m.

Answer: 15 m/s after traveling 38 m

**Example 2: braking distance**

Problem: A car moving at 24 m/s brakes with an acceleration of \(-5.0\ \text{m/s}^2\). How far does it go before it stops, and how long does that take?

1. Knowns: \(v_0 = 24\ \text{m/s}\), \(v = 0\), \(a = -5.0\ \text{m/s}^2\). No time given, so use \(v^2 = v_0^2 + 2a\Delta x\).

   $$
   0 = (24\ \text{m/s})^2 + 2(-5.0\ \text{m/s}^2)\,\Delta x
   $$
2. Solve for \(\Delta x\).

   $$
   \Delta x = \frac{576\ \text{m}^2/\text{s}^2}{10\ \text{m/s}^2} = 57.6\ \text{m}
   $$
3. Time, from \(v = v_0 + at\).

   $$
   t = \frac{0 - 24\ \text{m/s}}{-5.0\ \text{m/s}^2} = 4.8\ \text{s}
   $$

Answer: 58 m, in 4.8 s

**Example 3: dropping a ball**

Problem: A ball is dropped from a window 20.0 m above the ground. How long does it fall, and how fast is it going when it lands? Ignore air resistance.

1. Take down as positive this time so every value is positive: \(v_0 = 0\), \(a = 9.8\ \text{m/s}^2\), \(\Delta x = 20.0\ \text{m}\).
2. Time, from \(\Delta x = \tfrac{1}{2}at^2\).

   $$
   t = \sqrt{\frac{2\Delta x}{a}} = \sqrt{\frac{2(20.0\ \text{m})}{9.8\ \text{m/s}^2}} = 2.020\ \text{s}
   $$
3. Landing speed.

   $$
   v = at = (9.8\ \text{m/s}^2)(2.020\ \text{s}) = 19.80\ \text{m/s}
   $$

Answer: 2.02 s, landing at 19.8 m/s

**Example 4: throwing straight up**

Problem: A ball is thrown straight up at 15.0 m/s. How high does it go, and how long does it take to reach the top?

1. Up is positive: \(v_0 = 15.0\ \text{m/s}\), \(a = -9.8\ \text{m/s}^2\), and \(v = 0\) at the top.
2. Height, from the equation without time.

   $$
   \Delta x = \frac{v^2 - v_0^2}{2a} = \frac{0 - (15.0\ \text{m/s})^2}{2(-9.8\ \text{m/s}^2)} = 11.48\ \text{m}
   $$
3. Time to the top.

   $$
   t = \frac{v - v_0}{a} = \frac{0 - 15.0\ \text{m/s}}{-9.8\ \text{m/s}^2} = 1.531\ \text{s}
   $$

Answer: 11.5 m high, reached after 1.53 s

## Common mistakes and how to fix them

- **Mixing signs.** If up is positive, gravity is \(-9.8\ \text{m/s}^2\), even when the ball moves up. Fix: write your positive direction at the top of the page.
- **Using the equations when acceleration changes.** They only work for constant acceleration. Fix: split the motion into parts where \(a\) is constant.
- **Thinking acceleration is zero at the top.** Velocity is zero at the top, but gravity still acts, so \(a = -9.8\ \text{m/s}^2\).
- **Forgetting to square the time.** In \(\tfrac{1}{2}at^2\), square \(t\) before you multiply.
- **Mixing units.** Fix: convert km/h to m/s and minutes to seconds before you start.

**Practice problems**

1. A bike moving at 2.0 m/s speeds up at 1.5 m/s² for 4.0 s. What is its final speed?
   A. 6.0 m/s
   B. 8.0 m/s
   C. 14 m/s
   D. 3.5 m/s

   Answer: 8.0 m/s. \(v = v_0 + at = 2.0 + (1.5)(4.0) = 8.0\) m/s. 6.0 m/s forgets the starting speed.

2. A plane starts from rest, accelerates at 3.2 m/s², and needs 64 m/s to take off. What is the shortest runway it can use?

   Answer: 640 m. No time is given, so use \(v^2 = v_0^2 + 2a\Delta x\): \(\Delta x = \frac{64^2 - 0}{2(3.2)} = 640\) m.

3. Which equation should you use if a problem gives no time and asks for no time?
   A. \(v = v_0 + at\)
   B. \(\Delta x = v_0 t + \tfrac{1}{2}at^2\)
   C. \(v^2 = v_0^2 + 2a\Delta x\)
   D. \(\Delta x = \tfrac{1}{2}(v_0 + v)t\)

   Answer: \(v^2 = v_0^2 + 2a\Delta x\). It is the only one of the four with no \(t\) in it.

4. A stone is dropped from a bridge and hits the water 3.0 s later. How high is the bridge? Ignore air resistance.

   Answer: 44 m. \(\Delta x = \tfrac{1}{2}gt^2 = \tfrac{1}{2}(9.8)(3.0)^2 = 44.1\) m, which is 44 m to 2 significant figures.

5. A car slows from 30. m/s to 10. m/s in 4.0 s. What is its acceleration?
   A. -5.0 m/s²
   B. 5.0 m/s²
   C. -20 m/s²
   D. -2.5 m/s²

   Answer: -5.0 m/s². \(a = \frac{v - v_0}{t} = \frac{10 - 30}{4.0} = -5.0\) m/s². It is negative because the car is slowing down while moving in the positive direction.

## Frequently asked questions

### What is the difference between speed and velocity?

Speed is how fast something moves. Velocity is speed plus a direction, so it can be negative. A car going around a curve at a steady 20 m/s has constant speed but changing velocity. The kinematics equations use velocity, so the signs matter.

### Can I use these equations for objects thrown at an angle?

Yes, one direction at a time. Split the motion into horizontal and vertical parts and use the equations for each. Horizontally \(a = 0\), and vertically \(a = -9.8\ \text{m/s}^2\). See [projectile motion](https://duckyhelper.com/learn/physics/projectile-motion/).

### Why is g sometimes 9.81 or 10?

The real value near Earth's surface is about \(9.81\ \text{m/s}^2\) and changes a little from place to place. Many classes round to 9.8, and some use 10 to make the math quick. Use the value your teacher or test gives, and keep it the same throughout a problem.

## Sources

- [OpenStax College Physics 2e, 2.5 Motion Equations for Constant Acceleration in One Dimension](https://openstax.org/books/college-physics-2e/pages/2-5-motion-equations-for-constant-acceleration-in-one-dimension), accessed 2026-10-01
- [OpenStax College Physics 2e, 2.7 Falling Objects](https://openstax.org/books/college-physics-2e/pages/2-7-falling-objects), accessed 2026-10-01

## Related

- [Projectile motion](https://duckyhelper.com/learn/physics/projectile-motion/)
- [Newton's laws of motion](https://duckyhelper.com/learn/physics/newtons-laws-of-motion/)
- [Unit conversion and significant figures](https://duckyhelper.com/learn/physics/unit-conversion-and-significant-figures/)
- [Physics study guides](https://duckyhelper.com/learn/physics/)

## Try asking Ducky

- "I don't know which kinematics equation to use for number 3. Help me list what I know first."
- "Check my signs on this free fall problem. I said up is positive."
- "Give me a braking distance problem with different numbers and watch me solve it."

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