PHYS 210 Exam 1 Study Guide: Kinematics and Motion

Unit 1: Kinematics

Units and Conversions

Kinematics involves the study of motion without considering its causes. Accurate measurement requires understanding units and converting between them.

• SI Units: The standard units in physics are meters (m) for length, seconds (s) for time, and kilograms (kg) for mass.

• Unit Conversion: To convert between units, use conversion factors. For example, $1\ \text{km} = 1000\ \text{m}$.

• Example: Convert 5 km to meters: $5\ \text{km} \times 1000\ \text{m/km} = 5000\ \text{m}$.

Scalars and Vectors in One Dimension

Physical quantities are classified as scalars or vectors. Understanding the distinction is essential for analyzing motion.

• Scalar: A quantity with magnitude only (e.g., speed, distance).

• Vector: A quantity with both magnitude and direction (e.g., velocity, displacement).

• Example: Distance traveled (scalar) vs. displacement (vector).

Displacement, Velocity, and Acceleration

These are fundamental concepts describing motion in one dimension.

Displacement

• Definition: The change in position of an object; a vector quantity.

• Formula: $\Delta x = x_f - x_i$

• Example: If an object moves from $x_i = 2\ \text{m}$ to $x_f = 5\ \text{m}$, $\Delta x = 3\ \text{m}$.

Velocity

• Definition: The rate of change of displacement; a vector quantity.

• Average Velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$

• Instantaneous Velocity: $v = \frac{dx}{dt}$

• Speed: The magnitude of velocity; a scalar quantity.

• Position vs Time Plots: The slope of the position-time graph gives velocity.

• Example: If $\Delta x = 10\ \text{m}$ in $\Delta t = 2\ \text{s}$, $v_{avg} = 5\ \text{m/s}$.

Acceleration

• Definition: The rate of change of velocity; a vector quantity.

• Average Acceleration: $a_{avg} = \frac{\Delta v}{\Delta t}$

• Instantaneous Acceleration: $a = \frac{dv}{dt}$

• Velocity vs Time Plots: The slope of the velocity-time graph gives acceleration.

• Example: If $\Delta v = 6\ \text{m/s}$ in $\Delta t = 3\ \text{s}$, $a_{avg} = 2\ \text{m/s}^2$.

Representing Motion

Motion in one dimension can be represented using graphs and equations.

• Motion in 1D: Objects move along a straight line; position, velocity, and acceleration are described along this line.

• Kinematic Equations: For constant acceleration, the following equations apply:

• $v = v_0 + a t$

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

• $v^2 = v_0^2 + 2 a (x - x_0)$

• $x = x_0 + \frac{1}{2} (v_0 + v) t$

• Example: An object starts from rest ($v_0 = 0$) and accelerates at $2\ \text{m/s}^2$ for $3\ \text{s}$: $v = 0 + 2 \times 3 = 6\ \text{m/s}$.

Free Fall

Free fall describes the motion of objects under the influence of gravity alone.

• Acceleration due to Gravity: $g \approx 9.8\ \text{m/s}^2$ downward.

• Kinematic Equations for Free Fall: Replace $a$ with $-g$ (if upward is positive).

• Example: Dropping a ball from rest: $v = 0 + (-9.8) t$.

Vectors and Motion in Two Dimensions

Motion in two dimensions requires vector analysis and understanding projectile motion.

• Vectors: Quantities with both magnitude and direction; represented by arrows.

• Vector Addition: Use the Pythagorean theorem and trigonometry for components.

• Example: A displacement of 3 m east and 4 m north results in a total displacement of $\sqrt{3^2 + 4^2} = 5\ \text{m}$ northeast.

Projectile Motion

Projectile motion is a form of two-dimensional motion where an object moves under the influence of gravity.

• Horizontal Motion: Constant velocity; $x = v_{0x} t$

• Vertical Motion: Constant acceleration due to gravity; $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

• Time of Flight: Determined by vertical motion.

• Range: $R = \frac{v_0^2 \sin 2\theta}{g}$

• Maximum Height: $H = \frac{v_0^2 \sin^2 \theta}{2g}$

• Example: A projectile launched at $v_0 = 10\ \text{m/s}$ at $\theta = 45^\circ$.

Quantity	Formula	Description
Displacement	$\Delta x = x_f - x_i$	Change in position
Average Velocity	$v_{avg} = \frac{\Delta x}{\Delta t}$	Rate of change of displacement
Average Acceleration	$a_{avg} = \frac{\Delta v}{\Delta t}$	Rate of change of velocity
Range (Projectile)	$R = \frac{v_0^2 \sin 2\theta}{g}$	Horizontal distance traveled
Maximum Height (Projectile)	$H = \frac{v_0^2 \sin^2 \theta}{2g}$	Highest point reached

Additional info: Academic context and formulas have been expanded for completeness and clarity.