Where These Equations Come From

These aren't random formulas to memorize. Each equation comes from the graphs you just studied in the previous section.

From Graphs to Equations

Remember from section 1.2:

• The slope of a velocity-time graph is acceleration
• The area under a velocity-time graph is displacement

When acceleration is constant, the velocity-time graph is a straight line. That makes calculating slope and area straightforward: just geometry. Each kinematic equation captures one of these relationships.

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The Calculus Connection

If you know calculus, here's the deeper picture: velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity. Working backward (integrating) gives you the kinematic equations.

Don't know calculus? No problem. You can understand and use these equations perfectly well from the graphical approach. The key insight is that constant acceleration makes everything linear and predictable.

Let's look at each equation individually.

Equation 1: Velocity-Time Relationship

What It Tells You

This equation connects velocity, acceleration, and time. It answers questions like:

Note: Since this equation involves velocity (a vector), the result will reflect direction. If you want to know just the speed (magnitude of velocity), use

to represent the magnitude. When acceleration is in the same direction as velocity, the object speeds up; when opposite, it slows down.

• "If I start at 5 m/s and accelerate at 2 m/s² for 3 seconds, how fast am I going?"
• "How long does it take to go from 0 to 60 mph with constant acceleration?"

When to Use It

Use this equation when you don't need to know position. It's the simplest kinematic equation because it only involves three variables.

Missing variable: position (

), or distance ()

Where It Comes From

This is the definition of constant acceleration rearranged. Remember that acceleration is the change in velocity divided by time:

=

Δv

Δt

Since

= - , multiply both sides by and rearrange to get = + .

Equation 2: Position with Constant Acceleration

What It Tells You

This is the "master equation" for position. It tells you where an object ends up after accelerating for a certain time. Notice it has two parts:

• v ₀
  Δt
  (how far you'd go at constant initial velocity)
• ½
  a
  Δt ²
  (the extra distance from acceleration)

When to Use It

Use this when you know (or want to find) position or distance (

) and you have information about time and initial conditions.

Missing variable: final velocity (

)

Where It Comes From

This comes from the area under the velocity-time curve. For constant acceleration, that's a trapezoid. The area breaks down into:

• A rectangle:
  v ₀
  ×
  Δt
  (initial velocity contribution)
• A triangle: ½ × (
  a
  Δt
  ) ×
  Δt
  = ½
  a
  Δt ²
  (acceleration contribution)

Add your starting position

, and you get the full equation. When the problem asks for distance, the magnitude is used instead of displacement.

Equation 3: The Time-Independent Equation

What It Tells You

This equation relates velocity, acceleration, and displacement without involving time. It's incredibly useful when you don't know how long something took (or don't care).

Common questions it answers:

• "How fast is the car going after traveling 100 m with constant acceleration?" (Use with distance
  | d |
  or displacement
  Δx
  )
• "How far does a ball roll before stopping if it starts at 8 m/s and decelerates at -2 m/s²?" (Here
  | d |
  represents the stopping distance)

When to Use It

Use this when time is unknown or irrelevant. Perfect for problems where you're given velocities and distances, but time isn't mentioned.

Missing variable: time (

)

Where It Comes From

You can derive this by combining Equations 1 and 2 to eliminate time. Start with

= + , solve for , substitute into Equation 2, and simplify. The algebra works out to this clean relationship.

Equation 4: Average Velocity Form

What It Tells You

This equation connects position to average velocity. Remember from section 1.1 that average velocity is displacement divided by time? This equation says: displacement (or distance

) equals average velocity times time.

For constant acceleration, the average of initial and final velocities gives you the true average velocity:

+ 2 = , or for average speed

When to Use It

Use this when you know both initial and final velocities and want to find position (or vice versa). It's especially handy when you don't know acceleration directly.

Missing variable: acceleration (

)

Where It Comes From

This comes from the definition of average velocity. For constant acceleration, velocity changes linearly, so the average is simply the mean of initial and final values. Multiply by time to get displacement.

You can also derive it algebraically by combining Equations 1 and 2 to eliminate acceleration.

Choosing the Right Equation

The key to solving kinematic problems efficiently is picking the right equation. Here's a systematic approach:

Step-by-Step Strategy

1. List what you know: Write down all given values (position, velocity, acceleration, time). Remember that distance
   | d |
   is the magnitude and displacement
   Δx
   includes direction.
2. Identify what you want to find: What's the unknown variable? Is it
   | s |
   (speed) or
   v
   (velocity)?
3. Identify the missing variable: Which of the five kinematic variables (
   Δx
   ,
   v ₀
   ,
   v _f
   ,
   a
   ,
   Δt
   ) doesn't appear in the problem?
4. Choose the equation: Pick the equation that's missing the same variable you identified in step 3
5. Solve algebraically: Rearrange to isolate the unknown
6. Check your answer: Does the sign make sense? Is the magnitude reasonable?

Quick Reference: Missing Variables

Pro Tip

Sometimes you need to use multiple equations in sequence. Solve for an intermediate variable with one equation, then plug that result into another equation to find what you really want. Don't be afraid to chain them together!

Important Assumptions

These kinematic equations are powerful, but they're not universal. They come with specific conditions that must be met.

When These Equations Work

• Acceleration is constant: No changing acceleration mid-motion. If it varies, these equations break down.
• Motion is in a straight line: We're doing 1D kinematics here. Curved paths require vector components (coming in future chapters).
• Inertial reference frame: You're measuring from a non-accelerating perspective. (This is almost always the case in intro physics problems.)

When They Don't Work

If acceleration changes over time (like a rocket burning fuel, a car accelerating differently at various speeds, or air resistance causing complex deceleration), you can't use these equations directly.

For variable acceleration, you need calculus-based approaches: derivatives and integrals. That's beyond AP Physics 1, but it's good to know the limitation exists.

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Common Scenarios with Constant Acceleration

• Free fall near Earth's surface: Gravity provides constant
  a
  ≈ 9.8 m/s² downward (ignoring air resistance)
• Objects on frictionless surfaces: If net force is constant, acceleration is constant by Newton's Second Law
• Braking with constant force: Car brakes applying steady force leads to constant deceleration

These equations will be your workhorse throughout kinematics. Master them, and you'll solve motion problems with confidence.