Interactive physics simulator
Second Equation of Motion: s = ut + 1/2at²
The second equation of motion calculates displacement from initial velocity, constant acceleration, and time.
Second Equation of Motion Simulator
Use s = ut + 1/2at² for displacement when acceleration is constant. It does not directly calculate final velocity.
Live Result
- Initial velocity u
- 4 m/s
- Acceleration a
- 2 m/s²
- Time t
- 5 s
- Elapsed time
- 0 s
- Current displacement
- 0 m
- Final displacement s
- 45 m
- Current velocity
- 4 m/s
- Direction
- positive displacement
- Formula
- s = 4 x 5 + 1/2 x 2 x 5²
What is s = ut + 1/2at²?
The equation s = ut + 1/2at² calculates displacement during straight-line motion with constant acceleration. The ut part is displacement from initial velocity, and the 1/2at² part is extra displacement caused by acceleration.
Key Idea
The second equation of motion calculates displacement when initial velocity, acceleration, and time are known.
- It works for constant acceleration.
- It calculates displacement, not final velocity.
- The term ut shows displacement from initial velocity.
- The term 1/2at² shows extra displacement from acceleration.
- Displacement can be positive, negative, or zero.
- If acceleration is zero, the equation becomes s = ut.
Second Equation of Motion Formula
s = ut + 1/2at²
Displacement = Initial velocity x Time + 1/2 x Acceleration x Time²
s is displacement, u is initial velocity, a is acceleration, and t is time.
Use meters for displacement, m/s for velocity, m/s² for acceleration, and seconds for time.
Related forms
- s = ut when acceleration is zero.
- s = 1/2at² when the object starts from rest.
- v = u + at can support the motion, but it calculates final velocity.
When to Use This Formula
Use s = ut + 1/2at² when:
- Acceleration is constant.
- Initial velocity is known.
- Acceleration is known.
- Time is known.
- Displacement is required.
Do not use it alone when:
- You need final velocity only.
- Acceleration is changing.
- The motion is along a curved path and you need total distance.
- The time value is not known.
Real-life Example of s = ut + 1/2at²
A small rocket starts from rest and accelerates upward at 3 m/s² for 4 seconds.
u = 0 m/s, a = 3 m/s², t = 4 s.
s = 0 x 4 + 1/2 x 3 x 4² = 24 m.
- A ball rolling down a slope travels farther each second because acceleration adds extra displacement.
- A train capsule slowing down still moves forward, but covers less extra distance each second.
Solved Examples
An object starts from rest and accelerates at 2 m/s² for 5 s. Find displacement.
- s = ut + 1/2at²
- s = 0 x 5 + 1/2 x 2 x 5²
- s = 25 m
Answer: 25 m
A drone moves with initial velocity 4 m/s and acceleration 3 m/s² for 6 s. Find displacement.
- s = ut + 1/2at²
- s = 4 x 6 + 1/2 x 3 x 6²
- s = 24 + 54
- s = 78 m
Answer: 78 m
A ball moves at 20 m/s and slows with acceleration -2 m/s² for 4 s. Find displacement.
- s = ut + 1/2at²
- s = 20 x 4 + 1/2 x (-2) x 4²
- s = 80 - 16
- s = 64 m
Answer: 64 m
A particle moves at constant velocity 6 m/s for 8 s. Find displacement.
- a = 0, so s = ut
- s = 6 x 8
- s = 48 m
Answer: 48 m
Common Mistakes
- Using the formula when acceleration is not constant.
- Forgetting that time must be in seconds.
- Forgetting to square the time in the 1/2at² term.
- Leaving out the 1/2 in 1/2at².
- Confusing displacement with final velocity.
- Ignoring the sign of acceleration.
- Mixing meters, kilometers, seconds, and hours without conversion.
- Thinking displacement and total path distance are always the same.
Quick Summary
- Second equation of motion: s = ut + 1/2at².
- It calculates displacement.
- It works when acceleration is constant.
- u means initial velocity.
- a means acceleration.
- t means time.
- The 1/2at² term comes from acceleration.
- Use v = u + at if final velocity is required.
Practice Questions
1. An object starts from rest and accelerates at 4 m/s² for 3 s. Find displacement.
s = 0 x 3 + 1/2 x 4 x 3² = 18 m.
2. A drone has u = 5 m/s, a = 2 m/s², and t = 4 s. Find s.
s = 5 x 4 + 1/2 x 2 x 4² = 20 + 16 = 36 m.
3. A ball has u = 12 m/s, a = -3 m/s², and t = 2 s. Find s.
s = 12 x 2 + 1/2 x (-3) x 2² = 24 - 6 = 18 m.
4. If acceleration is zero, what does s = ut + 1/2at² become?
It becomes s = ut because 1/2at² is zero.
5. Does this equation directly calculate final velocity?
No. It directly calculates displacement. Use v = u + at for final velocity.
6. When is s = ut + 1/2at² valid?
It is valid when acceleration is constant.
FAQ
Frequently Asked Questions
What is the second equation of motion?
The second equation of motion is s = ut + 1/2at². It calculates displacement when initial velocity, acceleration, and time are known.
What does s mean in s = ut + 1/2at²?
s means displacement, or change in position, for straight-line motion with constant acceleration.
What does u mean in the second equation of motion?
u means initial velocity.
What does a mean in the second equation of motion?
a means constant acceleration.
What does t mean in the second equation of motion?
t means time, measured in seconds for SI calculations.
When can I use s = ut + 1/2at²?
Use it when acceleration is constant and you need displacement.
Does s = ut + 1/2at² calculate final velocity?
No. It directly calculates displacement. Final velocity is found with v = u + at.
Can displacement be negative in this equation?
Yes. Displacement can be negative if the motion is in the negative direction.
What happens if acceleration is zero?
If acceleration is zero, the equation becomes s = ut.
Why is there a t squared in the equation?
The t² term appears because acceleration changes velocity over time, so the extra displacement from acceleration grows faster as time increases.
What is the unit of displacement?
The SI unit of displacement is meter, written as m.
Is this equation valid for changing acceleration?
No. The equation s = ut + 1/2at² is for constant acceleration only.