AP Physics C: Mechanics - Unit 1: Kinematics

1. Fundamentals of Motion and Vectors

Definitions & Conceptual Basics

Kinematics is the branch of mechanics describing the motion of objects without reference to the forces causing the motion. In AP Physics C, unlike lower-level physics courses, we define motion variables using calculus and vector notation.

  • Scalar Quantities: Described by magnitude only (e.g., distance, speed, time).
  • Vector Quantities: Described by both magnitude and direction (e.g., displacement, velocity, acceleration).
Vector Notation and Components

In two or three dimensions, vectors are often decomposed into components along the Cartesian axes ($x, y, z$). We use unit vectors $\hat{i}, \hat{j}, \hat{k}$ to represent directions along the $x, y,$ and $z$ axes respectively.

For a position vector $\vec{r}$:
r⃗=xi^+yj^+zk^\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}

Key Vector Operations:

  • Magnitude: $|\vec{A}| = \sqrt{Ax^2 + Ay^2}$
  • Direction (Angle): $\theta = \tan^{-1}\left(\frac{Ay}{Ax}\right)$ (Be mindful of the quadrant!)
  • Addition: $\vec{R} = \vec{A} + \vec{B} = (Ax + Bx)\hat{i} + (Ay + By)\hat{j}$

Vector Decomposition Diagram

2. Motion in One Dimension: The Calculus Approach

In AP Physics C, kinematic variables are functions of time. You must be comfortable moving between position ($x$), velocity ($v$), and acceleration ($a$) using derivatives and integrals.

Position, Displacement, and Distance
  • Position ($x(t)$): The coordinate location of a particle at time $t$.
  • Displacement ($\Delta x$): The vector change in position.
    Δx=x<em>f−x</em>i=∫<em>t</em>itfv(t) dt\Delta x = x<em>f - x</em>i = \int<em>{t</em>i}^{t_f} v(t) \, dt
  • Distance: The total scalar path length. This is the integral of speed.
    d=∫<em>t</em>itf∣v(t)∣ dtd = \int<em>{t</em>i}^{t_f} |v(t)| \, dt
Instantaneous vs. Average Velocity
  • Average Velocity: depends only on endpoints.
    vavg=ΔxΔtv_{avg} = \frac{\Delta x}{\Delta t}
  • Instantaneous Velocity: The limit as time approaches zero; the derivative of position.
    v(t)=dxdtv(t) = \frac{dx}{dt}

Example Problem:
An object moves such that its position is $x(t) = 4t^2 - 3t + 2$. Find its instantaneous velocity at $t=2$s.
Solution:
$v(t) = \frac{d}{dt}(4t^2 - 3t + 2) = 8t - 3$.
At $t=2$: $v(2) = 8(2) - 3 = 13 \text{ m/s}$.

Instantaneous vs. Average Acceleration
  • Average Acceleration:
    aavg=ΔvΔta_{avg} = \frac{\Delta v}{\Delta t}
  • Instantaneous Acceleration: The derivative of velocity (second derivative of position).
    a(t)=dvdt=d2xdt2a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2}
Moving "Backwards" with Integration

If you are given acceleration and asked for velocity, or velocity and asked for position, you must integrate. Don't forget the constant of integration ($+C$), which represents initial conditions ($v0$ or $x0$).

v(t)=v<em>0+∫</em>0ta(t′) dt′v(t) = v<em>0 + \int</em>{0}^{t} a(t') \, dt'
x(t)=x<em>0+∫</em>0tv(t′) dt′x(t) = x<em>0 + \int</em>{0}^{t} v(t') \, dt'

3. Uniformly Accelerated Motion (Constant Acceleration)

When acceleration is constant (constant magnitude and direction), the calculus definitions simplify into the "Big Five" algebraic equations. These are the workhorses for free-fall problems and simple dynamics.

The "Big Five" Kinematic Equations
EquationMissing Variableapplication
$vx = v{x0} + a_xt$$\Delta x$Velocity as a function of time
$x = x0 + v{x0}t + \frac{1}{2}a_xt^2$$v_f$Position as a function of time
$vx^2 = v{x0}^2 + 2ax(x - x0)$$t$Velocity as a function of position
$x = x0 + \frac{1}{2}(vx + v_{x0})t$$a$Average velocity displacement
$x = x0 + vxt - \frac{1}{2}a_xt^2$$v_0$Motion calculated from final velocity
Free Fall
  • Definition: An object moving solely under the influence of gravity.
  • Condition: $a_y = -g$ (where $g \approx 9.8 \text{ m/s}^2$).
  • Air Resistance: In Unit 1 problems, unless specified, ignore air resistance. (Drag forces are covered in Unit 2).

Common Trap: At the peak of a tossed object's flight, velocity is zero, but acceleration is still gravity ($-9.8 \text{ m/s}^2$). Acceleration does not vanish just because the object stops momentarily.

4. Graphical Analysis of Motion

Interpreting graphs is a critical skill for the AP exam. The relationship between graphs mirrors the calculus relationships.

Relationship Table
Graph TypeSlope RepresentsArea Under Curve Represents
Position vs. Time ($x-t$)Velocity ($v$)N/A (physically meaningless)
Velocity vs. Time ($v-t$)Acceleration ($a$)Displacement ($\Delta x$)
Acceleration vs. Time ($a-t$)Jerk (rate of change of $a$)Change in Velocity ($\Delta v$)

Stacked Kinematics Graphs

Visual Check:

  • If $x-t$ is a parabola (degree 2), $v-t$ is a straight line (degree 1), and $a-t$ is a horizontal line (degree 0).
  • Concavity: On an $x-t$ graph, concave up means positive acceleration ($a > 0$); concave down means negative acceleration ($a < 0$).

5. Motion in Two Dimensions

5.1 Projectile Motion

The key concept in 2D kinematics is the Independence of Perpendicular Motions. Motion in the $x$-direction is independent of motion in the $y$-direction. They are linked only by time ($t$).

Standard Analysis Setup:

AxisAccelerationVelocity EquationPosition Equation
Horizontal ($x$)$a_x = 0$$vx = v0 \cos\theta$ (Constant)$x = x0 + (v0 \cos\theta)t$
Vertical ($y$)$a_y = -g$$vy = v0 \sin\theta - gt$$y = y0 + (v0 \sin\theta)t - \frac{1}{2}gt^2$

Projectile Motion Diagram

The Range Equation (Special Case):
Only valid if the projectile lands at the same height it was launched:
R=v02sin⁡(2θ)gR = \frac{v_0^2 \sin(2\theta)}{g}
Maximum range occurs at $\theta = 45^\circ$.

5.2 Relative Motion

Velocity is relative to the frame of reference. We use standard subscript notation:

  • $\vec{v}_{AC}$: Velocity of object A relative to C.
  • $\vec{v}_{AB}$: Velocity of object A relative to B.
  • $\vec{v}_{BC}$: Velocity of object B relative to C.

Vector Addition Rule:
v⃗<em>AC=v⃗</em>AB+v⃗BC\vec{v}<em>{AC} = \vec{v}</em>{AB} + \vec{v}_{BC}

mnemonic: The inner subscripts "match" and cancel out (B).

Example: A boat crosses a river.

  • $\vec{v}_{BW}$: Velocity of Boat relative to Water (engine speed).
  • $\vec{v}_{WE}$: Velocity of Water relative to Earth (river current).
  • $\vec{v}{BE} = \vec{v}{BW} + \vec{v}_{WE}$: Resultant velocity of Boat relative to Earth.

6. Common Mistakes & Pitfalls

  1. Confusing Distance and Displacement:

    • Running a lap around a 400m track: Distance = 400m, Displacement = 0.
    • Integrals: $\int v dt$ is displacement; $\int |v| dt$ is magnitude of distance.

  2. Sign Errors in Free Fall:

    • Always define your coordinate system first (usually Up = positive). If Up is positive, $a = -9.8$. If Down is positive, $a = +9.8$.
    • Make sure explicit velocities match signs (e.g., if throwing a ball down, $v_0$ is negative).

  3. Misinterpreting Graphs:

    • Thinking the intersection of two lines on a $v-t$ graph means the cars are colliding. (It just means they have the same speed; collisions happen when position lines intersect).

  4. Mixing Axes in Projectiles:

    • Using $v_y$ to calculate horizontal distance, or using $g$ in the $x$-equation. Keep your x and y tables completely separate until you solve for $t$.