Topic 4.2 Notes – Straight-Line Motion: Connecting Position, Velocity, and Acceleration

1. Position, Velocity, Acceleration, and Speed

In rectilinear motion, a particle moves along a straight line, usually the $x$-axis. Everything depends on time $t$.

Position

• $x(t)$ = position function
• Tells you where the particle is at time $t$
• Units: meters, feet, etc.

If $x(t) = 5$, that means the particle is 5 units to the right of the origin at that time.

Velocity

Velocity is the rate of change of position.

$$v(t) = x'(t)$$

• Signed quantity
  • $v(t) > 0$ → moving right
  • $v(t) < 0$ → moving left
  • $v(t) = 0$ → particle is at rest (at that instant)
• Units: distance per time (m/s, ft/s)

Velocity tells you direction and speed combined.

Speed

$$\text{Speed} = |v(t)|$$

• Always nonnegative
• Ignores direction

Students mix this up a lot. If velocity is $-4$, speed is $4$. The AP will absolutely test that difference.

Acceleration

Acceleration is the rate of change of velocity.

$$a(t) = v'(t) = x''(t)$$

• Units: distance per time²
• Tells you how velocity is changing

Chain of ideas:

$$x(t) \rightarrow v(t) \rightarrow a(t)$$

Differentiate once to get velocity. Differentiate again to get acceleration.

2. How to Find and Interpret Each Quantity

Everything depends on what you’re given.

If You’re Given $x(t)$ (position)

1. Differentiate once → $v(t)$
2. Differentiate again → $a(t)$
3. Plug in a time if they want a number

Example:

Let $x(t) = t^3 - 6t^2 + 9t$

$$v(t) = 3t^2 - 12t + 9$$

$$a(t) = 6t - 12$$

If they ask for acceleration at $t=4$, you substitute into $a(t)$.

On FRQs, don’t skip writing the derivative step. Points are often attached to it.

If You’re Given $v(t)$

• Differentiate → acceleration
• Integrate → position (use initial position if given)

If You’re Given $a(t)$

• Integrate → velocity (use initial velocity)
• Integrate again → position

That “integrate twice” situation shows up often on calculator FRQs.

Interpreting Signs

Velocity

• Positive → right
• Negative → left

Acceleration

• Positive → velocity increasing
• Negative → velocity decreasing
• Zero → constant velocity

Careful: acceleration does not tell you direction of motion. It tells you how velocity changes.

3. Speeding Up vs Slowing Down

This is one of the most tested ideas in motion problems.

The particle is:

• Speeding up when velocity and acceleration have the same sign
• Slowing down when they have opposite signs

Why this works:

• Acceleration changes velocity.
• If it pushes in the same direction velocity is already going, speed increases.
• If it pushes against it, speed decreases.

On tests, they love giving you a velocity graph and asking where the particle is speeding up. You check where velocity and the slope of the velocity graph have the same sign.

4. Common AP Question Types

When is the particle at rest?

Solve:

$$v(t) = 0$$

Not $a(t)=0$. That mistake costs points every year.

When is it moving right or left?

• Moving right → $v(t) > 0$
• Moving left → $v(t) < 0$

Usually you solve $v(t)=0$ and do a sign chart.

Constant Motion Situations

• If $a(t) = 0$, velocity is constant.
• If velocity is constant and nonzero, position is linear.
• If velocity is constantly zero, the particle isn’t moving at all.

Those connections between derivatives and function shapes matter a lot.