Kinematics and Projectile Motion: Study Notes for PHYSICS 115

Kinematics: Motion with Constant Acceleration

Introduction to Kinematic Equations

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. For motion with constant acceleration, a set of primary equations allows us to predict the position and velocity of an object at any time.

• Displacement Equation: The position of an object after time t is given by: $x_f = x_i + v_i t + \frac{1}{2} a t^2$ where $x_i$ is the initial position, $v_i$ is the initial velocity, $a$ is the constant acceleration, and $t$ is time elapsed.

• Velocity Equation: The velocity of an object after time t is: $v_f = v_i + a t$

• Component Form: For two-dimensional motion, these equations apply separately to the x (horizontal) and y (vertical) directions: $v_{fx} = v_{ix} + a_x t$ $x_f = x_i + v_{ix} t + \frac{1}{2} a_x t^2$ $v_{fy} = v_{iy} + a_y t$ $y_f = y_i + v_{iy} t + \frac{1}{2} a_y t^2$

Example: Calculating the final position and velocity of a car accelerating at $2\,\mathrm{m/s^2}$ for $5$ seconds from rest.

Tabular Summary of Kinematic Variables

Purpose: Organizing Variables for Problem Solving

When solving kinematics problems, it is helpful to organize known and unknown variables for each direction. The following table summarizes the variables and equations for both x and y directions:

x-direction	y-direction
$x_i$ = initial x-position	$y_i$ = initial y-position
$x_f$ = final x-position	$y_f$ = final y-position
$v_{ix}$ = initial x-velocity	$v_{iy}$ = initial y-velocity
$v_{fx}$ = final x-velocity	$v_{fy}$ = final y-velocity
$a_x$ = x-acceleration	$a_y$ = y-acceleration
$t$ = time	$t$ = time
$v_{fx} = v_{ix} + a_x t$ $x_f = x_i + v_{ix} t + \frac{1}{2} a_x t^2$	$v_{fy} = v_{iy} + a_y t$ $y_f = y_i + v_{iy} t + \frac{1}{2} a_y t^2$

Projectile Motion

Introduction to Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and moves under the influence of gravity alone (neglecting air resistance). The motion can be analyzed by separating it into horizontal and vertical components.

• Vertical Acceleration: The acceleration in the vertical direction is always $9.8\,\mathrm{m/s^2}$ downward, due to gravity.

• Horizontal Acceleration: The acceleration in the horizontal direction is zero ($a_x = 0$), meaning horizontal velocity remains constant.

• Initial Velocity Conditions:

  • If an object rolls off a horizontal surface, its initial vertical velocity ($v_{iy}$) is zero.

  • If an object is dropped, $v_{iy} = 0$.

  • If an object is thrown, $v_{iy} \neq 0$.

Example: A ball rolls horizontally off a table 1.20 m high and lands 1.47 m away. To find its initial speed, use the time to fall vertically and the horizontal distance traveled.

Projectile Motion: Tabular Analysis

For a ball rolling off a horizontal table, the following table summarizes the initial and final conditions:

x-direction	y-direction
$x_i$	$y_i$
$x_f$	$y_f$
$v_{ix} \neq 0$	$v_{iy} = 0$
$v_{fx} = v_{ix}$	$v_{fy} \neq v_{fi}$
$a_x = 0$	$a_y = 9.8\,\mathrm{m/s^2}$ Downward
$t$	$t$

Projectile Launched at an Angle

When a projectile is launched at an angle, its initial velocity must be resolved into horizontal and vertical components using trigonometry:

• $v_{ix} = v_i \cos \theta$

• $v_{iy} = v_i \sin \theta$

The following table summarizes the variables for a projectile shot upward at an angle:

x-direction	y-direction
$x_i$	$y_i$
$x_f$	$y_f$
$v_{ix}$	$v_{iy}$
$v_{fx}$	$v_{fy}$
$a_x$	$a_y$
$t$	$t$

Example: A projectile is shot upward at an angle of $23^\circ$ with a speed of $3\,\mathrm{m/s}$. To find how far away it lands, resolve the initial velocity into components and use the kinematic equations for each direction.

Worked Example: Ball Rolling Off a Table

Problem Statement

A small ball rolls horizontally off the edge of a tabletop that is 1.20 m high. It strikes the floor at a point 1.47 m horizontally away from the edge of the table. How fast was the ball moving when it left the table? Neglect air resistance.

• Step 1: Calculate the time to fall using vertical motion: $y_f = y_i + v_{iy} t + \frac{1}{2} a_y t^2$ Since $v_{iy} = 0$, $y_f = y_i + \frac{1}{2} a_y t^2$ Solve for $t$ using $y_i = 1.20\,\mathrm{m}$, $y_f = 0$, $a_y = -9.8\,\mathrm{m/s^2}$.

• Step 2: Use horizontal distance to find initial speed: $x_f = x_i + v_{ix} t$ $v_{ix} = \frac{x_f - x_i}{t}$ $x_f - x_i = 1.47\,\mathrm{m}$

Application: This method is used to analyze any projectile motion where the initial vertical velocity is zero.

Additional info: The notes also reference course logistics and online homework systems (MasteringPhysics), but the main academic content is focused on introductory kinematics and projectile motion.