PHYS1403: Practice Problems and Concepts in 1D Kinematics and Measurement

Introduction to Physics and Measurement

What is Physics?

Physics is the branch of science concerned with the study of matter, energy, and the fundamental forces of nature. It seeks to understand the laws governing the universe, from the smallest particles to the largest structures.

• Key Areas: Mechanics, thermodynamics, electromagnetism, optics, quantum physics, and relativity.

• Applications: Engineering, technology, medicine, and environmental science.

Units and Unit Conversions

Physical quantities are measured using standard units. The International System of Units (SI) is the most widely used system.

• Base SI Units: meter (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for temperature, mole (mol) for amount of substance, candela (cd) for luminous intensity.

• Unit Conversion: To convert between units, multiply by appropriate conversion factors (e.g., 1 km = 1000 m).

• Example: Convert 5 km to meters: $5~\text{km} \times \frac{1000~\text{m}}{1~\text{km}} = 5000~\text{m}$

Uncertainties and Significant Figures

Measurements in physics are never exact; they always have some degree of uncertainty. Significant figures reflect the precision of a measurement.

• Significant Figures: The digits in a measurement that are known with certainty plus one estimated digit.

• Rules: Leading zeros are not significant; zeros between nonzero digits are significant; trailing zeros after a decimal point are significant.

• Example: The number 0.04 has 1 significant figure.

• Calculations: When adding/subtracting, the result should have as many decimal places as the least precise measurement. When multiplying/dividing, the result should have as many significant figures as the measurement with the fewest significant figures.

• Example Calculation: $\frac{72.0 - 15.60}{0.982 + 4.3}$ (Apply significant figure rules to the result.)

Dimensional Analysis

Dimensional analysis is a method to check the consistency of equations by comparing the dimensions on both sides.

• Basic Dimensions: Length [L], Mass [M], Time [T]

• Example: Is the equation $x = vt - 2a/t^2$ dimensionally correct, where $x$ is position, $v$ is velocity, $a$ is acceleration, and $t$ is time?

• Analysis: $[x] = L$, $[v] = L/T$, $[a] = L/T^2$, $[t] = T$

• Check: $vt$ has dimension $L/T \times T = L$; $a/t^2$ has dimension $L/T^2 \div T^2 = L/T^4$ (so $2a/t^2$ is not dimensionally consistent with $x$).

Problem Solving Strategy

• Identify knowns and unknowns.

• Draw diagrams if applicable.

• Write relevant equations.

• Solve algebraically before substituting numbers.

• Check units and significant figures in the final answer.

Motion in One Dimension (1D Kinematics)

Position, Displacement, and Distance

Describing motion in one dimension involves understanding how an object's position changes over time.

• Position (x): The location of an object along a straight line.

• Displacement (Δx): The change in position; a vector quantity.

• Distance: The total length of the path traveled; a scalar quantity.

• Example: If an object moves 8 m east, then 3 m west, the distance is 11 m, and the displacement is 5 m east.

Velocity and Speed

Velocity and speed describe how fast an object moves and in what direction.

• Average Velocity ($\bar{v}$): $\bar{v} = \frac{\Delta x}{\Delta t}$

• Instantaneous Velocity: The velocity at a specific instant; the slope of the x-t graph at a point.

• Speed: The magnitude of velocity; always positive.

• Example: For a round trip, the average speed is always greater than zero, but the average velocity can be zero if the displacement is zero.

Acceleration

Acceleration is the rate at which velocity changes with time.

• Average Acceleration ($\bar{a}$): $\bar{a} = \frac{\Delta v}{\Delta t}$

• Instantaneous Acceleration: The acceleration at a specific instant; the slope of the v-t graph at a point.

• Direction: Acceleration can be positive or negative, depending on whether the object is speeding up or slowing down in a given direction.

• Example: If an object is moving left with negative velocity and slowing down, its acceleration is positive (opposite to velocity).

Graphical Analysis (x-t and v-t Graphs)

Graphs are useful for visualizing motion.

• x-t Graph: Position vs. time; the slope gives velocity.

• v-t Graph: Velocity vs. time; the slope gives acceleration, and the area under the curve gives displacement.

• Example: A straight, upward-sloping x-t graph corresponds to a constant positive velocity; the matching v-t graph is a horizontal line above the time axis.

Equations of Motion (Constant Acceleration)

For motion with constant acceleration, the following kinematic equations apply:

• $v = v_0 + at$

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

• $v^2 = v_0^2 + 2a(x - x_0)$

Where:

• $x$ = final position

• $x_0$ = initial position

• $v$ = final velocity

• $v_0$ = initial velocity

• $a$ = acceleration

• $t$ = time

Table: Kinematic Equations and Included Quantities

The following table summarizes which variables are included in each kinematic equation:

Equation	t	x	x_0	v	v_0	a
$v = v_0 + a t$	✓			✓	✓	✓
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	✓	✓	✓		✓	✓
$v^2 = v_0^2 + 2a(x - x_0)$		✓	✓	✓	✓	✓

Vectors and Vector Components

Vector Components and Magnitude

Vectors have both magnitude and direction. They can be broken into components along the x and y axes.

• Component Form: $\vec{A} = A_x \hat{i} + A_y \hat{j}$

• Magnitude: $|\vec{A}| = \sqrt{A_x^2 + A_y^2}$

• Direction: $\theta = \arctan\left(\frac{A_y}{A_x}\right)$

• Example: If $A_x = -8$ units and $A_y = 6$ units, the angle with the positive x-axis is $\arctan(6/-8)$.

Vector Addition

Vectors are added component-wise to find the resultant vector.

• Resultant: $\vec{R} = \vec{F} + \vec{G} + \vec{H}$

• Direction: Determined by the signs and magnitudes of the components.

• Example: If all components are negative, the resultant points in the negative direction of both axes.

Force Components

Forces can be resolved into x and y components using trigonometry.

• Given: Force $F$ with magnitude $F$ at angle $\theta$ from the x-axis.

• x-component: $F_x = F \cos \theta$

• y-component: $F_y = F \sin \theta$

• Example: $F = 25.0~\text{N}$ at $30^\circ$ north of east: $F_x = 25.0 \cos 30^\circ$

Conceptual Questions and Problem Types

• Distance vs. Displacement: Distance is the total path length; displacement is the straight-line change in position.

• Average Speed vs. Average Velocity: Average speed is always positive; average velocity can be zero for a round trip.

• Acceleration Direction: If an object slows down, acceleration is opposite to velocity.

• Graph Interpretation: The slope of an x-t graph gives velocity; the slope of a v-t graph gives acceleration.

• Significant Figures in Calculations: Always round the final answer to the correct number of significant figures based on the input data.

Example Problems

• Problem: A car accelerates from 15 m/s to 25 m/s in 5 s. What distance does it travel?

• Solution: Use $x = x_0 + v_0 t + \frac{1}{2} a t^2$ or $v = v_0 + at$ to find $a$, then $x$.

• Problem: A cheetah slows from 20.0 m/s to rest in 5.00 s. How far does it travel?

• Solution: Use $x = v_0 t + \frac{1}{2} a t^2$ with $a = (v - v_0)/t$.

Additional info: Some context and explanations have been expanded for clarity and completeness, as the original material was in question format and brief notes.