S - Displacement

Symbol: \(s\), \(\Delta x\), or \(d\)

Unit: meters (m)

Definition: The change in position from the starting point to the final point. It's a vector quantity with both magnitude and direction.

Note: Displacement differs from distance - it considers direction and can be negative.

U - Initial Velocity

Symbol: \(u\), \(v_0\), or \(v_i\)

Unit: meters per second (m/s)

Definition: The velocity of the object at the beginning of the time interval (at \(t = 0\)).

Note: If the object starts from rest, \(u = 0\).

V - Final Velocity

Symbol: \(v\) or \(v_f\)

Unit: meters per second (m/s)

Definition: The velocity of the object at the end of the time interval.

Note: This is what you're often trying to find in problems.

A - Acceleration

Symbol: \(a\)

Unit: meters per second squared (m/s²)

Definition: The rate of change of velocity. Must be constant for these equations to apply.

Note: Negative acceleration means slowing down or accelerating in the negative direction.

T - Time

Symbol: \(t\) or \(\Delta t\)

Unit: seconds (s)

Definition: The duration of motion - the time interval between initial and final states.

Note: Time is always positive in these equations.

\[ v = u + at \]

Variables Used: \(v\), \(u\), \(a\), \(t\)

Missing Variable: Displacement (\(s\))

Use When: You need to find final velocity and don't know or need displacement.

Physical Meaning: Final velocity equals initial velocity plus the change in velocity due to acceleration over time.

\[ s = ut + \frac{1}{2}at^2 \]

Variables Used: \(s\), \(u\), \(a\), \(t\)

Missing Variable: Final velocity (\(v\))

Use When: You need to find displacement and don't know final velocity.

Physical Meaning: Displacement equals the distance traveled at initial velocity plus the additional distance from acceleration.

\[ s = vt - \frac{1}{2}at^2 \]

Variables Used: \(s\), \(v\), \(a\), \(t\)

Missing Variable: Initial velocity (\(u\))

Use When: You know final velocity but not initial velocity.

Physical Meaning: Alternative form working backward from final velocity.

\[ v^2 = u^2 + 2as \]

Variables Used: \(v\), \(u\), \(a\), \(s\)

Missing Variable: Time (\(t\))

Use When: Time is unknown and not needed - very useful for free fall problems!

Physical Meaning: Relates velocity change to displacement and acceleration without involving time.

\[ s = \frac{(u + v)}{2} \times t \]

Variables Used: \(s\), \(u\), \(v\), \(t\)

Missing Variable: Acceleration (\(a\))

Use When: You know both velocities but not acceleration.

Physical Meaning: Displacement equals average velocity multiplied by time (only valid for constant acceleration).

\[ a = \frac{v - u}{t} \]

\[ s = \frac{u + v}{2} \times t \]

Substitute \(v = u + at\):

\[ s = \frac{u + (u + at)}{2} \times t = \frac{2u + at}{2} \times t \]

\[ s = ut + \frac{1}{2}at^2 \]

\[ t = \frac{v - u}{a} \]

Substitute into \(s = ut + \frac{1}{2}at^2\):

After algebraic manipulation:

\[ v^2 = u^2 + 2as \]

• Acceleration: \(a = g = 9.81 \text{ m/s}^2\) (downward)

• If dropped from rest: \(u = 0\)

• Distance fallen: \(s = \frac{1}{2}gt^2\)

• Velocity after falling height \(h\): \(v = \sqrt{2gh}\)

• Time to fall height \(h\): \(t = \sqrt{\frac{2h}{g}}\)

Horizontal Motion (x-direction):

• Acceleration: \(a_x = 0\) (no air resistance)

• Velocity: \(v_x = u_x = \text{constant}\)

• Displacement: \(x = u_x t\)

Vertical Motion (y-direction):

• Acceleration: \(a_y = -g = -9.81 \text{ m/s}^2\)

• Use all kinematic equations with \(a = -g\)

• \(v_y = u_y - gt\) and \(y = u_y t - \frac{1}{2}gt^2\)

\[ v = at \]

\[ s = \frac{1}{2}at^2 \]

\[ v^2 = 2as \]

\[ s = \frac{v}{2} \times t \]

Step 1: List Known Values

Write down all five SUVAT variables. Fill in what you know from the problem. Mark unknown values with "?".

Step 2: Identify Target

Determine which variable you need to find. This narrows down which equations might work.

Step 3: Choose Equation

Select the equation that contains your target variable and doesn't include the variable you don't know.

Step 4: Substitute & Solve

Plug in your known values and solve algebraically. Watch your signs (positive/negative)!

Step 5: Check Units

Verify that all units are consistent (SI units preferred). Convert if necessary before calculating.

Step 6: Verify Answer

Does your answer make physical sense? Is the magnitude reasonable? Is the sign correct?

❌ Sign Errors

Problem: Mixing up positive and negative directions.

Solution: Define a positive direction at the start. Stick to it throughout. Upward is usually positive.

❌ Unit Inconsistencies

Problem: Mixing km/h with m/s, or using cm instead of m.

Solution: Convert everything to SI units (m, s, m/s, m/s²) before calculating.

❌ Using Wrong Equation

Problem: Choosing an equation that includes an unknown variable you don't need.

Solution: List all five variables first. Pick the equation without the "extra" unknown.

❌ Forgetting Squared Terms

Problem: Writing \(v = u^2 + 2as\) instead of \(v^2 = u^2 + 2as\).

Solution: Remember to square root when solving for velocity in this equation.

❌ Assuming Constant Velocity

Problem: Using \(s = vt\) when there's acceleration.

Solution: \(s = vt\) only works when \(a = 0\). Otherwise use kinematic equations.

❌ Confusing Distance and Displacement

Problem: Using total path length instead of straight-line displacement.

Solution: Remember \(s\) is displacement (vector), not total distance traveled.

Problem:

A car accelerates from rest at 2.5 m/s² for 8 seconds. How far does it travel, and what is its final velocity?

Step 1: List Known Values

• \(u = 0\) m/s (starts from rest)

• \(a = 2.5\) m/s²

• \(t = 8\) s

• \(v = ?\) (need to find)

• \(s = ?\) (need to find)

Step 2: Find Final Velocity

Use: \(v = u + at\) (we have \(u\), \(a\), and \(t\))

\[ v = 0 + (2.5)(8) = 20 \text{ m/s} \]

Step 3: Find Displacement

Use: \(s = ut + \frac{1}{2}at^2\) (we have \(u\), \(a\), and \(t\))

\[ s = (0)(8) + \frac{1}{2}(2.5)(8)^2 \]

\[ s = 0 + 1.25 \times 64 = 80 \text{ m} \]

✓ Answer:

The car travels 80 meters and reaches a final velocity of 20 m/s.

About the Author

Adam

Co-Founder @ RevisionTown

Math Expert specializing in various curricula including IB, AP, GCSE, IGCSE, and more