Kinematics and Solving Motion Problems: Study Notes

Kinematics: Equations of Motion

Independent Equations and Unknowns

Kinematics is the study of motion without considering its causes. When solving kinematics problems, it is crucial to ensure that the number of independent equations matches or exceeds the number of unknowns. This allows for a complete solution to the problem.

• Key Point 1: If you have more unknowns than independent equations, the problem cannot be solved.

• Key Point 2: There are only two independent equations of motion for each dimension (e.g., x or y).

Origin of Motion Equations

The fundamental equations of motion are derived from the definitions of average velocity and average acceleration. These equations form the basis for solving constant acceleration problems in one dimension.

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

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

• Position equation (using average velocity): $x_f = x_i + \frac{1}{2}(v_i + v_f)t$

• Velocity equation (using average acceleration): $v_f = v_i + a t$

Recombination of Kinematics Equations

All other kinematics equations are derived from the two primary equations above, especially under constant acceleration conditions.

• For constant acceleration:

• $v_{avg} = \frac{v_f + v_i}{2}$

• $x_f = x_i + v_i t + \frac{1}{2} a t^2$

Example: If an object starts at $x_i$ with initial velocity $v_i$ and constant acceleration $a$, its position after time $t$ is:

• $x_f = x_i + v_i t + \frac{1}{2} a t^2$

Solving Motion Problems

Systematic Approach to Kinematics Problems

To solve kinematics problems efficiently, follow a structured approach:

• Step 1: Choose and clearly state a coordinate system.

• Step 2: Assign symbols to all quantities and make a list of knowns and unknowns.

• Step 3: Fill in numbers (or symbols) with correct signs, based on the chosen coordinate system.

• Step 4: Use the appropriate kinematics equations to solve for the unknowns.

Typical variables:

• $x_i$ = initial position

• $x_f$ = final position

• $v_i$ = initial velocity

• $v_f$ = final velocity

• $a$ = acceleration

• $t$ = time interval

Vector Components and Signs

Vector quantities such as displacement, velocity, and acceleration have direction. The sign (+ or -) indicates direction along the chosen axis.

• Key Point: The meaning of the sign depends on the coordinate system you choose and state explicitly.

Checking Your Solution

After solving, always verify your answer:

• Check that the answer is reasonable.

• Ensure units are consistent (dimensional analysis).

Worked Example: Elevator Problem

Problem Statement

An elevator moves downward and slows at a steady rate of $2\ \mathrm{m/s^2}$. It starts at $10\ \mathrm{m/s}$ (downward) and comes to a stop. How far does the elevator travel during this time?

• Knowns: $v_i = 10\ \mathrm{m/s}$ (downward), $v_f = 0$, $a = -2\ \mathrm{m/s^2}$ (negative since slowing down), $x_i = 0$ (if starting from origin)

• Unknown: $x_f$ (distance traveled)

Solution outline:

• Use $v_f = v_i + a t$ to solve for $t$.

• Use $x_f = x_i + v_i t + \frac{1}{2} a t^2$ to find $x_f$.

Additional info: This is a classic constant acceleration problem, often used to illustrate the use of kinematics equations.

"Catch Up" Problems

Relative Motion and Tables

"Catch up" problems involve two objects moving along the same axis, often with different initial velocities and/or accelerations. The goal is to determine when and where one object overtakes the other.

• Step 1: Make a table listing initial and final positions, velocities, accelerations, and time for each object.

• Step 2: Choose a coordinate system and state it explicitly.

• Step 3: Write the kinematics equations for each object:

Quantity	Bob (B)	Alice (A)
Initial Position ($x_{Bi}$, $x_{Ai}$)	$x_{Bi}$	$x_{Ai}$
Final Position ($x_{Bf}$, $x_{Af}$)	$x_{Bf}$	$x_{Af}$
Initial Velocity ($v_{Bi}$, $v_{Ai}$)	$v_{Bi}$	$v_{Ai}$
Final Velocity ($v_{Bf}$, $v_{Af}$)	$v_{Bf}$	$v_{Af}$
Acceleration ($a_B$, $a_A$)	$a_B$	$a_A$
Time ($t$)	$t$	$t$

• Position equation: $x_f = x_i + \frac{1}{2}(v_i + v_f)t$

• Velocity equation: $v_f = v_i + a t$

Example: If Bob starts from rest and Alice moves at constant velocity, set $x_{Bf} = x_{Af}$ and solve for $t$ to find when Bob catches up.

Additional info: Tables help organize information and clarify which variables belong to which object.