Worked Examples

Example 1: Finding Final Velocity (1st Equation)

A car starts from rest and accelerates at 4 m/s² for 6 seconds. What is its final velocity?

Given: u = 0 m/s, a = 4 m/s², t = 6 s

Formula: v = u + at

Substitution: v = 0 + (4)(6) = 24 m/s

Answer: The car’s final velocity is 24 m/s.

Example 2: Finding Displacement (2nd Equation)

A ball is thrown upward with an initial velocity of 20 m/s. How far does it travel in 3 seconds? (Take a = −10 m/s²)

Given: u = 20 m/s, a = −10 m/s², t = 3 s

Formula: s = ut + ½at²

Substitution: s = (20)(3) + ½(−10)(3²) = 60 + ½(−10)(9) = 60 − 45 = 15 m

Answer: The ball travels 15 m upward from its launch point after 3 seconds.

Example 3: Finding Acceleration Without Time (3rd Equation)

A cyclist moving at 12 m/s brakes and comes to a complete stop over a distance of 36 m. What is the deceleration?

Given: u = 12 m/s, v = 0 m/s, s = 36 m

Formula: v² = u² + 2as

Substitution: 0² = 12² + 2a(36) → 0 = 144 + 72a → 72a = −144 → a = −2 m/s²

Answer: The deceleration is 2 m/s² (magnitude).

Common Mistakes to Avoid

• Ignoring the sign of acceleration. Students often treat deceleration as a positive value. Acceleration is a vector — if an object is slowing down, acceleration must be entered as a negative number. Failing to do so gives a physically impossible answer.
• Using inconsistent units. Mixing km/h for speed with seconds for time produces wrong results. Always convert all values to the same unit system (SI: m, s, m/s, m/s²) before substituting into any formula.
• Assuming initial velocity is always zero. Many students set u = 0 by default. Only do this when the problem explicitly states the object starts from rest. If the object is already moving, use the given initial velocity.
• Choosing the wrong equation when time is unknown. If the problem gives displacement and velocity but not time, the 3rd equation (v² = u² + 2as) is the correct choice. Trying to force the 1st or 2nd equation wastes time and introduces errors.
• Forgetting the ½ factor in the 2nd equation. A common arithmetic slip is writing s = ut + at² instead of s = ut + ½at². The factor of one-half comes from the integration of velocity and must always be included.

Real-World Applications

• Vehicle braking distance: Traffic engineers use v² = u² + 2as to calculate the minimum stopping distance for cars at highway speeds. This directly informs safe following distances and speed limit decisions on roads.
• Projectile motion in sports: When a basketball player shoots the ball, coaches and analysts apply s = ut + ½at² to model the vertical position of the ball over time, helping optimize launch angle and release speed.
• Rocket and spacecraft launches: During the powered ascent phase of a rocket, mission planners use v = u + at to estimate the velocity gained over a burn period, ensuring the craft reaches the required orbital speed.
• Free-fall and skydiving: Skydivers in free fall before deploying a parachute accelerate under gravity. The 2nd equation lets instructors calculate how far a diver falls in a given number of seconds, which is critical for altitude and parachute deployment timing.

Frequently Asked Questions

What are the three equations of motion?

The three equations of motion are v = u + at, s = ut + ½at², and v² = u² + 2as. They describe the relationship between initial velocity, final velocity, acceleration, time, and displacement for an object moving with uniform (constant) acceleration. All three assume acceleration does not change during the motion.

When should I use the 3rd equation instead of the others?

Use v² = u² + 2as whenever the problem gives you displacement and velocities but does not mention time, or when finding time would require an extra step. It is especially useful in free-fall and braking problems where time is neither given nor asked for.

Do these equations work for deceleration?

Yes. Simply enter acceleration as a negative value when the object is slowing down. The equations handle deceleration correctly as long as signs are applied consistently throughout the calculation.

Can these equations be used for vertical motion under gravity?

Yes. For objects in free fall or thrown vertically, substitute a = −9.8 m/s² (or −10 m/s² for quick estimates). Make sure to define a positive direction — usually upward — and keep all signs consistent with that choice.

What does uniform acceleration mean?

Uniform acceleration means the acceleration stays constant throughout the motion — it does not increase or decrease. These three equations are only valid under this condition. Real-world situations like air resistance cause non-uniform acceleration, which requires more advanced methods.

Why does the 2nd equation have a ½ in it?

The ½ arises from calculus: displacement is the integral of velocity over time. Since velocity increases linearly as v = u + at, the area under the velocity-time graph is a trapezoid, and its area formula naturally produces the ½ factor. It is not optional — omitting it gives a result that is too large by exactly half the acceleration term.