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

Introduction to Physics and Measurement

What is Physics?

Physics is the branch of science that studies matter, energy, and the fundamental forces of nature. It seeks to understand the behavior of the universe through observation, experimentation, and mathematical modeling.

• 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 inch = 2.54 cm).

Uncertainties and Significant Figures

All measurements have some degree of uncertainty. Significant figures reflect the precision of a measurement.

• Significant Figures: Digits in a number that carry meaning contributing to its precision.

• Rules: All nonzero digits are significant; zeros between nonzero digits are significant; leading zeros are not significant; trailing zeros in a decimal number are significant.

• Example: The number 0.04 has 1 significant figure.

Dimensional Analysis

Dimensional analysis checks the consistency of equations by comparing the dimensions on both sides.

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

• Example: For $x = vt - 2a/t^2$, check if dimensions match: $[x] = L$, $[v] = L/T$, $[a] = L/T^2$, $[t] = T$.

• Application: Ensures equations are physically meaningful.

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

Describes how objects move along a straight line.

• Position (x): Location of an object relative to an origin.

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

• Distance: Total length of path traveled; a scalar quantity.

• Example: If an object moves 8 m east, then 3 m west: Distance = 11 m, Displacement = 5 m east.

Velocity and Speed

Describes how fast and in what direction an object moves.

• 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: Scalar magnitude of velocity; $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

Acceleration

Acceleration is the rate of change of velocity with respect to 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.

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

Graphs are useful for visualizing motion.

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

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

• Example: If an object moves rightward, slows to a stop, then moves leftward, the velocity is zero at the stop, but acceleration is not necessarily zero.

Equations of Motion (Constant Acceleration)

For motion with constant acceleration, the following 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: Variables in Kinematic Equations

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 in Physics

Vector Components and Magnitude

Vectors have both magnitude and direction. They can be broken into components along coordinate axes.

• Components: For a vector $\vec{A}$, $A_x$ and $A_y$ are its x and y components.

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

• Direction: $\theta = \arctan\left(\frac{A_y}{A_x}\right)$ (measured from the positive x-axis)

• Example: If $A_x = -8$, $A_y = 6$, $|\vec{A}| = 10$, $\theta = \arctan(-6/8)$

Vector Addition

Vectors are added component-wise.

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

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

• Example: If all components are negative, the resultant vector points in the negative direction.

Force Components

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

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

• x-component: $F_x = F \cos\theta$

• y-component: $F_y = F \sin\theta$

• Example: $F = 25.0$ N at $37^\circ$ north of east: $F_x = 25.0 \cos 37^\circ$, $F_y = 25.0 \sin 37^\circ$

Conceptual Questions and Problem Solving

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

• Average Velocity for Round Trip: For a round trip, displacement is zero, so average velocity is zero, but average speed is not.

• Significant Figures in Calculations: The result should have as many significant figures as the least precise measurement.

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

Example Problems

• Car Acceleration: A car accelerates from 15 m/s to 25 m/s in 5 s with $a = 2.0$ m/s$^2$. Distance traveled: $x = v_0 t + \frac{1}{2} a t^2$.

• Cheetah Deceleration: A cheetah slows from 20.0 m/s to rest in 5.00 s. Distance: $x = v_0 t + \frac{1}{2} a t^2$ (with $a$ negative).

• Duck Acceleration: A duck covers 2.0 m in the first second from rest. Find acceleration and distance after 5 s using $x = \frac{1}{2} a t^2$.

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