Kinematics · ⁨運動学⁩

Learning Objective 1.1.A: Describe a scalar or vector quantity using magnitude and direction, as appropriate.

• 1.1.A.1 Scalars are quantities described by magnitude only; vectors are quantities described by both magnitude and direction.
• 1.1.A.2 Vectors can be visually modeled as arrows with appropriate direction and lengths proportional to their magnitude.
• 1.1.A.3 Distance and speed are examples of scalar quantities, while position, displacement, velocity, and acceleration are examples of vector quantities.
• 1.1.A.4 Vectors can be expressed in unit vector notation or as a magnitude and a direction.
  • 1.1.A.4.i Unit vector notation can be used to represent vectors as the sum of their constituent components in the $x$-, $y$-, and $z$-directions, denoted by $\hat{i}$, $\hat{j}$, and $\hat{k}$, respectively.
    • Equation: $\vec{r} = \left( A\hat{i} + B\hat{j} + C\hat{k} \right)$
  • 1.1.A.4.ii The position vector of a point is given by $\vec{r}$, and the unit vector in the direction of the position vector is denoted $\hat{r}$.
  • 1.1.A.4.iii A resultant vector is the vector sum of the addend vectors' components.
    • Equation: $\vec{C} = \vec{A} + \vec{B}$
    • Equation: $\vec{C} = \left( A_x + B_x \right)\hat{i} + \left( A_y + B_y \right)\hat{j}$
• 1.1.A.5 In a given one-dimensional coordinate system, opposite directions are denoted by opposite signs.

Learning Objective 1.2.A: Describe a change in an object's position.

• 1.2.A.1 When using the object model, the size, shape, and internal configuration are ignored. The object may be treated as a single point with extensive properties such as mass and charge.
• 1.2.A.2 Displacement is the change in an object's position.
  • Equation: $\Delta x = x - x_0$

Learning Objective 1.2.B: Describe the average velocity and acceleration of an object.

• 1.2.B.1 Averages of velocity and acceleration are calculated considering the initial and final states of an object over an interval of time.
• 1.2.B.2 Average velocity is the displacement of an object divided by the interval of time in which that displacement occurs.
  • Equation: $\vec{v}_{\text{avg}} = \dfrac{\Delta \vec{x}}{\Delta t}$
• 1.2.B.3 Average acceleration is the change in velocity divided by the interval of time in which that change in velocity occurs.
  • Equation: $\vec{a}_{\text{avg}} = \dfrac{\Delta \vec{v}}{\Delta t}$
• 1.2.B.4 An object is accelerating if either the magnitude and/or direction of the object's velocity are changing.
• 1.2.B.5 Calculating average velocity or average acceleration over a very small time interval yields a value that is very close to the instantaneous velocity or instantaneous acceleration.

Learning Objective 1.2.C: Describe the instantaneous position, velocity, and acceleration of an object as a function of time.

• 1.2.C.1 As the time interval used to calculate the average value of a quantity approaches zero, the average value of that quantity approaches the value of the quantity at that instant, called the instantaneous value.
  • 1.2.C.1.i Instantaneous velocity is the rate of change of the object's position, which is equal to the derivative of position with respect to time.
    • Equation: $\vec{v} = \dfrac{d\vec{r}}{dt}$
    • Equation: $v_x = \dfrac{dx}{dt}$
  • 1.2.C.1.ii Instantaneous acceleration is the rate of change of the object's velocity, which is equal to the derivative of velocity with respect to time.
    • Equation: $\vec{a} = \dfrac{d\vec{v}}{dt}$
    • Equation: $a_x = \dfrac{dv_x}{dt}$
• 1.2.C.2 Time-dependent functions and instantaneous values of position, velocity, and acceleration can be determined using differentiation and integration.

Learning Objective 1.3.A: Describe the position, velocity, and acceleration of an object using representations of that object's motion.

• 1.3.A.1 Motion can be represented by motion diagrams, figures, graphs, equations, and narrative descriptions.
• 1.3.A.2 For constant acceleration, three kinematic equations can be used to describe instantaneous linear motion in one dimension:
  • Equation: $v_x = v_{x0} + a_x t$
  • Equation: $x = x_0 + v_{x0} t + \dfrac{1}{2} a_x t^2$
  • Equation: $v_x^2 = v_{x0}^2 + 2a_x \left( x - x_0 \right)$
  • Note: The equations above are written to indicate motion in the $x$-direction, but these equations can be used in any single dimension as appropriate.
• 1.3.A.3 Near the surface of Earth, the vertical acceleration caused by the force of gravity is downward, constant, and has a measured value approximately equal to
  • Equation: $a_g = g \approx 10 \ \text{m/s}^2$.
• 1.3.A.4 Graphs of position, velocity, and acceleration as functions of time can be used to find the relationships between those quantities.
  • 1.3.A.4.i An object's instantaneous velocity is the rate of change of the object's position, which is equal to the slope of a line tangent to a point on a graph of the object's position as a function of time.
    • Equation: $v_x = \dfrac{dx}{dt}$
  • 1.3.A.4.ii An object's instantaneous acceleration is the rate of change of the object's velocity, which is equal to the slope of a line tangent to a point on a graph of the object's velocity as a function of time.
    • Equation: $a_x = \dfrac{dv_x}{dt}$
  • 1.3.A.4.iii The displacement of an object during a time interval is equal to the area under the curve of a graph of the object's velocity as a function of time (i.e., the area bounded by the function and the horizontal axis for the appropriate interval).
    • Equation: $\Delta x = \displaystyle\int_{t_1}^{t_2} v_x(t) \ dt$
  • 1.3.A.4.iv The change in velocity of an object during a time interval is equal to the area under the curve of a graph of the acceleration of the object as a function of time.
    • Equation: $\Delta v_x = \displaystyle\int_{t_1}^{t_2} a_x(t) \ dt$

Boundary statement: AP Physics C: Mechanics and AP Physics C: Electricity and Magnetism expects that for all situations in which a numerical quantity is required for g, the value $g \approx 10 \ \text{m/s}^2$ will be used. However, students will not be penalized for correctly using the more precise commonly accepted values of $g = 9.81 \ \text{m/s}^2$ or $g = 9.8 \ \text{m/s}^2$.

Learning Objective 1.4.A: Describe the reference frame of a given observer.

• 1.4.A.1 The choice of reference frame will determine the direction and magnitude of quantities measured by an observer in that reference frame.

Learning Objective 1.4.B: Describe the motion of objects as measured by observers in different inertial reference frames.

• 1.4.B.1 Measurements from a given reference frame may be converted to measurements from another reference frame.
• 1.4.B.2 The observed velocity of an object results from the combination of the object's velocity and the velocity of the observer's reference frame.
  • 1.4.B.2.i Combining the motion of an object and the motion of an observer in a given reference frame involves the addition or subtraction of vectors.
  • 1.4.B.2.ii The acceleration of any object is the same as measured from all inertial reference frames.

Boundary statement: Unless otherwise stated, the frame of reference of any problem may be assumed to be inertial.

Learning Objective 1.5.A: Describe the motion of an object moving in two or three dimensions.

• 1.5.A.1 Motion in two or three dimensions can be analyzed using one-dimensional kinematic relationships if the motion is separated into components.
• 1.5.A.2 Velocity and acceleration may be different in each dimension and may be nonuniform.
• 1.5.A.3 Motion in one dimension may be changed without causing a change in a perpendicular dimension.
• 1.5.A.4 Projectile motion is a special case of two-dimensional motion that has zero acceleration in one dimension and constant, nonzero acceleration in the second dimension.

Boundary statement: AP Physics C: Mechanics only expects students to quantitatively analyze the motion of an object in two dimensions. AP Physics C: Electricity and Magnetism expects students to also qualitatively describe the motion of a particle in three dimensions.