Physics Equations of Motion- Formulas and Problem Solving
What Are the Equations of Motion?
Physics equations of motion are mathematical relationships that describe how an object's position, velocity, and acceleration change over time. That's it. No fancy metaphors needed.
These equations only work for objects moving with constant acceleration in a straight line. If acceleration is changing, these formulas break down and you need calculus. Keep that in mind before you start plugging numbers.
The three equations are commonly called SUVAT equations. The acronym comes from the variables they use:
- S – displacement
- U – initial velocity
- V – final velocity
- A – acceleration
- T – time
The Three Equations of Motion
Equation 1: v = u + at
This connects final velocity to initial velocity, acceleration, and time. Use it when you need to find velocity but don't know displacement.
Equation 2: s = ut + ½at²
This gives you displacement based on initial velocity, acceleration, and time. No final velocity required.
Equation 3: v² = u² + 2as
This relates velocities to displacement without involving time. Useful when time isn't given in the problem.
Equation 4: s = ½(u + v)t
This calculates displacement using average velocity and time. Clean and straightforward.
Most textbooks teach three equations. The fourth is just as useful, so memorize all four or know how to derive them from first principles.
Units You Need to Know
Before solving anything, check your units. Mess these up and your answer will be wrong every time.
| Variable | SI Unit | Common Mistakes |
|---|---|---|
| Displacement (s) | Meters (m) | Mixing km with m |
| Velocity (u, v) | Meters per second (m/s) | Using km/h without converting |
| Acceleration (a) | Meters per second² (m/s²) | Gravity is 9.8, not 10 (usually) |
| Time (t) | Seconds (s) | Hours mixed with seconds |
Which Equation Do You Use?
Here's the real problem most students face: they don't know which equation to pick. The answer is simpler than your textbook makes it seem.
Look at what information the problem gives you. Then look at what it's asking for. The equation you need is the one containing those variables.
- Problem gives: u, a, t → asks for v → use v = u + at
- Problem gives: u, a, t → asks for s → use s = ut + ½at²
- Problem gives: u, a, s → asks for v → use v² = u² + 2as
- Problem gives: u, v, t → asks for s → use s = ½(u + v)t
That's the entire selection process. No magic. Just matching.
How to Solve Motion Problems
Step 1: List What You Know
Write down all five SUVAT variables. Put a question mark next to the one you're solving for. Leave the rest blank if they're not given.
Step 2: Pick the Right Equation
Choose the equation that contains only your known variables and the one you're solving for. If an equation has an unknown variable besides your target, it's useless to you.
Step 3: Plug In the Numbers
Substitute your values directly. Keep units consistent. Convert everything to meters, seconds, and m/s before you touch the equation.
Step 4: Solve
Rearrange algebraically if needed, then calculate. Double-check your arithmetic. Most errors happen in this step, not in selecting the equation.
Worked Example
Problem: A car accelerates from rest (u = 0) at 4 m/s² for 6 seconds. What distance does it cover?
Step 1: List knowns
u = 0 m/s
a = 4 m/s²
t = 6 s
s = ?
Step 2: Pick equation
We have u, a, and t. We need s. The equation with u, a, t, and s is s = ut + ½at².
Step 3: Plug in
s = (0)(6) + ½(4)(6)²
s = 0 + ½(4)(36)
s = 2 × 36
s = 72 meters
Done. No fluff needed.
Common Mistakes That Ruin Your Answers
- Ignoring the direction of acceleration. Acceleration can be negative. If velocity and acceleration point opposite ways, your object is slowing down, not speeding up.
- Using g = 10 when the problem says 9.8. Teachers notice this. It costs you marks.
- Forgetting that "from rest" means u = 0. You'd think this is obvious. It isn't, based on how often it happens.
- Mixing up initial and final velocity. Label clearly. Swapping u and v gives you a negative answer that makes no sense.
- Not converting km/h to m/s. Multiply by 3.6 to go from km/h to m/s. Divide by 3.6 to go the other way.
Comparing the Equations
| Equation | Use When | Avoid When |
|---|---|---|
| v = u + at | Finding final velocity; time is known | Time isn't given |
| s = ut + ½at² | Finding displacement; no final velocity | v is given but t is missing |
| v² = u² + 2as | Time is not given or needed | Time is the target variable |
| s = ½(u + v)t | Both velocities are known; need displacement | Only one velocity is given |
Vertical Motion: When Gravity Changes Things
For objects thrown upward or falling, acceleration is replaced by g = 9.8 m/s² (downward). The same SUVAT equations apply. The difference is that one of your quantities will usually be negative.
A ball thrown upward with initial velocity 20 m/s:
u = 20 m/s (positive, upward)
a = -9.8 m/s² (negative, downward)
At maximum height, v = 0
The symmetry works like this: an object thrown upward with speed v reaches the same height as an object falling from that height with speed v. Use this to check your answers.
Deriving the Equations (For Those Who Want Proof)
You don't need to memorize everything blindly. The first equation comes directly from the definition of acceleration:
a = (v - u) / t
Rearrange: v = u + at ✓
From there, combine v = u + at with average velocity concepts to derive the rest. If your instructor asks you to derive these on a test, you can.
Bottom Line
The equations of motion are tools. You pick the right tool for the given information, plug in numbers, and solve. That's the entire process.
Stop overcomplicating it. Stop reading paragraphs that tell you motion is "the foundation of classical mechanics" or some garbage. You need the formulas, the right one for each situation, and practice solving actual problems.
Do that and you'll solve any motion problem they throw at you.