Study Guide: Kinematics in One, Two, and Three Dimensions

Chapter 2: Kinematics in One Dimension

Velocity and Acceleration – Average and Instantaneous

Kinematics is the study of motion without considering its causes. Two fundamental quantities in kinematics are velocity and acceleration, each of which can be described as average or instantaneous values.

• Average Velocity (\( \vec{v}_{avg} \)): The total displacement divided by the total time interval.

• Instantaneous Velocity (\( \vec{v} \)): The velocity of an object at a specific instant, defined as the derivative of position with respect to time.

• Average Acceleration (\( \vec{a}_{avg} \)): The change in velocity divided by the time interval over which the change occurs.

• Instantaneous Acceleration (\( \vec{a} \)): The acceleration at a specific instant, defined as the derivative of velocity with respect to time.

Key Equations:

• Average velocity: $\vec{v}_{avg} = \frac{\Delta \vec{x}}{\Delta t}$

• Instantaneous velocity: $\vec{v} = \frac{d\vec{x}}{dt}$

• Average acceleration: $\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}$

• Instantaneous acceleration: $\vec{a} = \frac{d\vec{v}}{dt}$

Example: If a car moves from 0 m to 100 m in 5 s, its average velocity is $\frac{100\,\mathrm{m}}{5\,\mathrm{s}} = 20\,\mathrm{m/s}$.

Higher-Order Derivatives: Jerk

In real-world scenarios, acceleration may not be constant. The rate of change of acceleration with respect to time is called jerk.

• Jerk (\( j \)): Defined as $j = \frac{da}{dt}$, with SI units of $\mathrm{m/s^3}$.

Example: If acceleration increases from $2\,\mathrm{m/s^2}$ to $6\,\mathrm{m/s^2}$ in 2 seconds, the average jerk is $\frac{6-2}{2} = 2\,\mathrm{m/s^3}$.

Kinematics and Gravity

Objects moving under the influence of gravity (with negligible air resistance) experience a constant acceleration directed downward, denoted as $g$.

• Acceleration due to gravity: $g \approx 9.8\,\mathrm{m/s^2}$ downward near Earth's surface.

• 1-D Kinematic Equations (constant acceleration):

  • $v = v_0 + at$

  • $x = x_0 + v_0 t + \frac{1}{2} a t^2$

  • $v^2 = v_0^2 + 2a(x - x_0)$

• Sign Conventions: Positive and negative signs indicate direction. For vertical motion, upward is often positive and downward is negative (or vice versa, as defined in the problem).

Example: A ball dropped from rest falls with $a = -g$. After 2 s, $v = 0 + (-9.8)(2) = -19.6\,\mathrm{m/s}$.

SI Units Review

• Displacement: meters (m)

• Velocity: meters per second (m/s)

• Acceleration: meters per second squared (m/s2)

• Jerk: meters per second cubed (m/s3)

Chapter 3: Kinematics in Two or Three Dimensions

Vector Kinematics and Vector Calculus

In two or three dimensions, position, velocity, and acceleration are described as vectors. Calculations often use unit vector notation (\( \hat{i}, \hat{j}, \hat{k} \)).

• Adding Vectors: Vectors are added component-wise. For example, $\vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k}$.

• Position Vector: $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$

• Velocity Vector: $\vec{v}(t) = \frac{d\vec{r}}{dt}$

• Acceleration Vector: $\vec{a}(t) = \frac{d\vec{v}}{dt}$

Example: If $\vec{r}(t) = (2t)\hat{i} + (3t^2)\hat{j}$, then $\vec{v}(t) = 2\hat{i} + 6t\hat{j}$ and $\vec{a}(t) = 6\hat{j}$.

2-D Projectile Motion

Projectile motion describes the path of an object moving under the influence of gravity alone (no air resistance). The path is a parabola.

• Shape of Path: Parabolic trajectory.

• Acceleration: Gravity acts only on the vertical component ($a_y = -g$), while the horizontal component ($a_x$) is zero.

• Equations:

  • Horizontal: $x = v_{0x} t$

  • Vertical: $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

• Horizontal Velocity: Remains constant throughout the flight ($v_x = v_{0x}$).

Example: A ball is launched horizontally at $10\,\mathrm{m/s}$ from a $20\,\mathrm{m}$ high cliff. Time to hit the ground: $t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{40}{9.8}} \approx 2.02\,\mathrm{s}$.

Relative Velocity and Frames of Reference

A frame of reference is a coordinate system from which motion is observed and measured. Relative velocity describes how the velocity of an object appears to different observers.

• Relative Velocity Equation: $\vec{v}_{A/B} = \vec{v}_A - \vec{v}_B$

• Projectile Motion in Different Frames: The equations of 1-D kinematics are valid in any inertial frame (stationary or moving at constant velocity).

Example: If a boat moves east at $5\,\mathrm{m/s}$ relative to the water, and the water flows north at $3\,\mathrm{m/s}$ relative to the ground, the boat's velocity relative to the ground is $\sqrt{5^2 + 3^2} = 5.83\,\mathrm{m/s}$ at an angle $\theta = \tan^{-1}(3/5) \approx 31^{\circ}$ north of east.

SI Units Review (2-D and 3-D)

• Position: meters (m)

• Velocity: meters per second (m/s)

• Acceleration: meters per second squared (m/s2)

Summary Table: Key Kinematic Quantities

Quantity	Symbol	Definition	SI Unit
Displacement	\( \Delta x, \vec{r} \)	Change in position	m
Velocity (average)	\( \vec{v}_{avg} \)	$\frac{\Delta \vec{x}}{\Delta t}$	m/s
Velocity (instantaneous)	\( \vec{v} \)	$\frac{d\vec{x}}{dt}$	m/s
Acceleration (average)	\( \vec{a}_{avg} \)	$\frac{\Delta \vec{v}}{\Delta t}$	m/s2
Acceleration (instantaneous)	\( \vec{a} \)	$\frac{d\vec{v}}{dt}$	m/s2
Jerk	\( j \)	$\frac{da}{dt}$	m/s3

Additional info: Students are encouraged to review all assigned homework and example problems, as well as practice with additional textbook exercises for mastery of these concepts.