Equation Sheet for Exam 2: Motion, Forces, and Orbits

Motion in One and Two Dimensions

Basic Kinematic Equations

Kinematics describes the motion of objects without considering the causes of motion. The following equations are fundamental for analyzing motion in one and two dimensions.

• Displacement (s): The change in position of an object.

• Velocity (v): The rate of change of displacement.

• Acceleration (a): The rate of change of velocity.

Key Equations:

• Displacement with constant acceleration: $s = v_0 t + \frac{1}{2} a t^2$

• Final velocity: $v = v_0 + a t$

• Velocity squared: $v^2 = v_0^2 + 2 a s$

Example: A car accelerates from rest at $2\ \mathrm{m/s^2}$ for 5 seconds. Its displacement is $s = 0 + \frac{1}{2} (2)(5^2) = 25\ \mathrm{m}$.

Projectile Motion

Equations for Projectile Motion

Projectile motion involves two-dimensional motion under the influence of gravity. The horizontal and vertical motions are analyzed separately.

• Horizontal acceleration: $a_x = 0$

• Vertical acceleration: $a_y = g$ (where $g = 9.8\ \mathrm{m/s^2}$ downward)

• Horizontal displacement: $x = v_{0x} t$

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

Example: A ball is launched horizontally from a table with $v_{0x} = 3\ \mathrm{m/s}$. The time to hit the ground depends only on the height and gravity.

Ramp Motion

Inclined Plane Analysis

When analyzing motion on a ramp (inclined plane), the gravitational force is resolved into components parallel and perpendicular to the surface.

• Parallel component of gravity: $W_{\parallel} = mg \sin \theta$

• Perpendicular component of gravity: $W_{\perp} = mg \cos \theta$

• Acceleration down the ramp (no friction): $a = g \sin \theta$

Example: A block slides down a frictionless ramp at $30^\circ$. Its acceleration is $a = 9.8 \sin 30^\circ = 4.9\ \mathrm{m/s^2}$.

Newton's Second Law and Forces

Newton's Second Law

Newton's Second Law relates the net force acting on an object to its acceleration.

• Equation: $\sum \vec{F} = m \vec{a}$

• Weight: $W = mg$

Example: A 2 kg object under a net force of 10 N accelerates at $a = F/m = 5\ \mathrm{m/s^2}$.

Periodic Motion and Orbits

Period, Frequency, and Acceleration

Periodic motion describes objects that repeat their motion in regular intervals, such as planets in orbit or masses on springs.

• Period (T): The time for one complete cycle.

• Frequency (f): The number of cycles per second. $f = \frac{1}{T}$

• Centripetal acceleration: $a_c = \frac{v^2}{r}$

Example: A satellite orbits Earth with a period of 90 minutes. Its frequency is $f = 1/(90 \times 60) \approx 0.000185\ \mathrm{Hz}$.

Summary Table: Key Equations

Topic	Equation	Description
Displacement (constant a)	$s = v_0 t + \frac{1}{2} a t^2$	Position after time t
Final velocity	$v = v_0 + a t$	Velocity after time t
Projectile (horizontal)	$x = v_{0x} t$	Horizontal position
Projectile (vertical)	$y = v_{0y} t - \frac{1}{2} g t^2$	Vertical position
Ramp acceleration	$a = g \sin \theta$	Down a frictionless incline
Newton's Second Law	$\sum \vec{F} = m \vec{a}$	Net force and acceleration
Weight	$W = mg$	Force of gravity
Period-Frequency	$f = \frac{1}{T}$	Relationship between period and frequency
Centripetal acceleration	$a_c = \frac{v^2}{r}$	Acceleration for circular motion

Additional info: Some equations and context were inferred based on standard introductory physics curriculum and the provided headings.