Interactive physics simulator
Third Equation of Motion: v² = u² + 2as
The third equation of motion finds final velocity from initial velocity, acceleration, and displacement when acceleration is constant.
Third Equation of Motion Simulator
Use v² = u² + 2as when acceleration is constant and time is not known or not needed.
Live Result
- Initial velocity u
- 5 m/s
- Acceleration a
- 2 m/s²
- Displacement s
- 20 m
- Current displacement
- 0 m
- v² value
- 105 m²/s²
- Final velocity v
- 10.2 m/s
- Current velocity
- 5 m/s
- Status
- physically possible
- Formula
- v² = 5² + 2 x 2 x 20
What is v² = u² + 2as?
The equation v² = u² + 2as finds final velocity for straight-line motion with constant acceleration. It is especially useful when time is not given. The equation first gives v², so the final speed magnitude is found by taking the square root.
Key Idea
The third equation of motion connects velocity, acceleration, and displacement without using time.
- It works for constant acceleration.
- It calculates final velocity from displacement.
- It does not contain time.
- It uses displacement, not total path distance.
- Take the square root of v² to find speed magnitude.
- If u² + 2as is negative, the input values are not physically possible.
Third Equation of Motion Formula
v² = u² + 2as
Final velocity² = Initial velocity² + 2 x Acceleration x Displacement
v is final velocity, u is initial velocity, a is acceleration, and s is displacement.
Use m/s for velocity, m/s² for acceleration, and meters for displacement.
Rearranged forms
- v = √(u² + 2as) to find speed magnitude.
- a = (v² - u²) / 2s to find acceleration.
- s = (v² - u²) / 2a to find displacement.
When to Use This Formula
Use v² = u² + 2as when:
- Acceleration is constant.
- Initial velocity is known.
- Acceleration is known.
- Displacement is known.
- Final velocity is required.
- Time is not given or not needed.
Do not use it alone when:
- Acceleration is changing.
- You need time directly.
- You need total distance along a curved path.
- The direction context is missing for velocity sign.
Real-life Example of v² = u² + 2as
A small rocket starts from rest and accelerates upward at 2 m/s² through a displacement of 25 m.
u = 0 m/s, a = 2 m/s², s = 25 m.
v² = 0² + 2 x 2 x 25 = 100, so v = 10 m/s.
- A ball rolling down a slope can use this equation if the slope displacement is known.
- A train capsule braking can use it to find stopping speed or stopping distance.
Solved Examples
A small rocket starts from rest, accelerates at 2 m/s², and moves 25 m. Find final velocity.
- v² = u² + 2as
- v² = 0² + 2 x 2 x 25
- v² = 100
- v = 10 m/s
Answer: 10 m/s
A drone has u = 5 m/s, a = 3 m/s², and s = 12 m. Find final velocity.
- v² = 5² + 2 x 3 x 12
- v² = 25 + 72
- v² = 97
- v = √97 = 9.8 m/s
Answer: 9.8 m/s
A ball moving at 20 m/s slows with acceleration -4 m/s² over 30 m. Find final velocity.
- v² = 20² + 2 x (-4) x 30
- v² = 400 - 240
- v² = 160
- v = √160 = 12.6 m/s
Answer: 12.6 m/s
A particle has u = 8 m/s and stops after 16 m. Find acceleration.
- v² = u² + 2as
- 0² = 8² + 2 x a x 16
- -64 = 32a
- a = -2 m/s²
Answer: -2 m/s²
Common Mistakes
- Forgetting to square u and v.
- Forgetting to take the square root at the end.
- Using time even though this equation does not need time.
- Using distance along a curved path instead of straight-line displacement.
- Ignoring the sign of acceleration or displacement.
- Thinking a negative v² value is allowed for real motion.
- Using the equation when acceleration is not constant.
- Forgetting that square root gives magnitude; direction comes from context.
Quick Summary
- Third equation of motion: v² = u² + 2as.
- It connects velocity, acceleration, and displacement.
- It works for constant acceleration.
- It does not contain time.
- u means initial velocity.
- v means final velocity.
- a means acceleration.
- s means displacement.
Practice Questions
1. An object starts from rest and accelerates at 4 m/s² over 8 m. Find final velocity.
v² = 0² + 2 x 4 x 8 = 64, so v = 8 m/s.
2. A drone has u = 6 m/s, a = 2 m/s², and s = 10 m. Find v.
v² = 6² + 2 x 2 x 10 = 76, so v = √76 = 8.7 m/s.
3. A ball slows from 12 m/s to 4 m/s over 20 m. Find acceleration.
a = (v² - u²) / 2s = (16 - 144) / 40 = -3.2 m/s².
4. Does v² = u² + 2as need time?
No. It is useful when time is not known or not needed.
5. When is v² = u² + 2as valid?
It is valid for straight-line motion with constant acceleration.
6. What does a negative value of v² mean?
It means the chosen values are not physically possible for real motion to that displacement.
FAQ
Frequently Asked Questions
What is the third equation of motion?
The third equation of motion is v² = u² + 2as. It relates final velocity, initial velocity, acceleration, and displacement.
What does v mean in v² = u² + 2as?
v means final velocity. Because the equation gives v² first, take the square root to find the speed magnitude.
What does u mean in the third equation of motion?
u means initial velocity.
What does a mean in the third equation of motion?
a means constant acceleration.
What does s mean in the third equation of motion?
s means displacement in straight-line motion.
When should I use v² = u² + 2as?
Use it when acceleration is constant and time is not known or not needed.
Does the third equation of motion use time?
No. The third equation of motion does not contain time.
Can v² be negative?
For real motion, v² cannot be negative. If u² + 2as is negative, the chosen values are not physically possible for reaching that displacement.
Can acceleration be negative in this equation?
Yes. Negative acceleration is allowed and often represents slowing down in the chosen positive direction.
Is displacement the same as distance in this equation?
In this equation, s is displacement along the motion line. It is not always total path distance.
Is v always positive after taking the square root?
The square root gives the magnitude. Direction must be decided from the motion context.
Is this equation valid for changing acceleration?
No. The equation v² = u² + 2as is valid only when acceleration is constant.