Study Notes: Motion at a Constant Speed and Kinematics Fundamentals

Motion at a Constant Speed

Introduction to Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. The study of motion at a constant speed is foundational for understanding more complex motion in physics.

• Constant Speed: An object moves at a constant speed if it covers equal distances in equal intervals of time.

• Displacement vs. Distance: Distance is the total length of the path traveled, while displacement is the straight-line distance from the initial to the final position.

• Reference: Sections 5, 6 of Chapter 1 and Sections 1-3 of Chapter 2 of Giancoli's textbook.

Distances, Times, and Powers of Ten

Units and Conversions

Physics uses standardized units to measure quantities such as length, time, and mass. Understanding unit conversions and the use of powers of ten is essential for solving problems efficiently.

• Length Units:

  • 1 meter (m) = 100 centimeters (cm) = $10^3$ millimeters (mm) = $10^6$ micrometers ($ext{μm}$) = $10^9$ nanometers (nm)

  • 1 kilometer (km) = 1,000 meters (m)

  • 1 mile (mi) = 1.609 kilometers (km) = 1,609 meters (m)

  • 1 foot (ft) = 30.5 centimeters (cm) = 0.305 meters (m)

• Time Units:

  • 1 hour (h) = 60 minutes (min) = 3,600 seconds (s)

  • 1 day = 24 hours

  • 1 year = 365 days

• Powers of Ten:

  • 1 thousand = $10^3$

  • 1 million = $10^6$

  • 1 billion = $10^9$

  • 1 trillion = $10^{12}$

Examples and Applications

• Example: Converting speed from miles per hour to meters per second: $55.0 ext{ mi/h} imes 1.609 ext{ km/mi} imes 1000 ext{ m/km} / 3600 ext{ s/h} ightarrow 24.6 ext{ m/s}$

• Example: Calculating how many times a line of people (8 billion, each separated by 2m) would circle the Earth's equator (40,000 km): Total line length = $8 imes 10^9 imes 2 ext{ m} = 1.6 imes 10^{10} ext{ m}$ Equator length = $4 imes 10^7 ext{ m}$ Number of times = $1.6 imes 10^{10} / 4 imes 10^7 = 400$

Speed and Velocity

Definitions

Speed and velocity are fundamental concepts in kinematics. Speed is a scalar quantity, while velocity is a vector, including both magnitude and direction.

• Average Speed ($ar{v}$): $\bar{v} = \frac{l_{tot}}{t_{tot}}$ where $l_{tot}$ is the total distance traveled and $t_{tot}$ is the total time taken.

• Instantaneous Speed: The speed of an object at a specific moment in time, as shown by a speedometer.

• Constant Velocity: $x = x_0 + vt$ where $x_0$ is the initial position, $v$ is velocity, and $t$ is time.

Examples

• Example: Usain Bolt's 100m world record: $v = \frac{100 ext{ m}}{9.580 ext{ s}} \approx 10.44 ext{ m/s}$ Converting to mi/h: $10.44 ext{ m/s} \times \frac{1 ext{ mi}}{1609 ext{ m}} \times \frac{3600 ext{ s}}{1 ext{ h}} \approx 23.4 ext{ mi/h}$

• Example: Trip from New York to Boston (215 mi at 55 mi/h): $t = \frac{l}{v} = \frac{215 ext{ mi}}{55 ext{ mi/h}} \approx 3.9 ext{ h}$

Average Speed in Segmented Motion

Motion with Multiple Segments

When an object moves at different speeds over different segments, the average speed is not simply the arithmetic mean of the speeds. The total distance and total time must be considered.

• Example: Car travels 150 mi: first half at 45 mi/h, second half at 75 mi/h. Incorrect method: $(45 + 75)/2 = 60$ mi/h (arithmetic mean, not correct) Correct method: $t_1 = \frac{75 ext{ mi}}{45 ext{ mi/h}} \approx 1.67 ext{ h}$ $t_2 = \frac{75 ext{ mi}}{75 ext{ mi/h}} = 1 ext{ h}$ $t_{tot} = 2.67 ext{ h}$ $\bar{v} = \frac{150 ext{ mi}}{2.67 ext{ h}} \approx 56 ext{ mi/h}$

Graphical Representation of Motion

Position-Time Graphs

Graphs are useful for visualizing motion. A position-time graph for constant velocity is a straight line, while changes in direction or speed are shown by changes in slope.

• Equation: $x = x_0 + vt$

• Example: A body stands at $x = 2$ from $t = 2$ to $t = 6$, then moves in the opposite direction until $t = 8$, then stands still, then moves again.

Oscillatory Motion: The Pendulum

Pendulum Period and Speed

The period of a simple pendulum depends on its length and the acceleration due to gravity. The speed of the pendulum bob varies during its swing.

• Pendulum Period: $T = 2\pi \sqrt{\frac{l}{g}}$ where $l$ is the length and $g$ is the acceleration due to gravity.

• Average Speed in a 90° Swing: The bob travels a quarter of a circle of radius $l$. $\bar{v} = \frac{\pi l}{2\pi \sqrt{l/g}} = \frac{1}{2} \sqrt{g l}$ For $l = 2$ m, $g = 9.8$ m/s$^2$: $\bar{v} = 0.5 \sqrt{9.8 \times 2} \approx 2.2$ m/s

Calculus in Kinematics (Advanced)

Derivatives and Integrals

Calculus provides precise definitions for instantaneous velocity and total distance traveled.

• Instantaneous Velocity: $v_A = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$

• Example: If $x = k t^2$, then $\frac{dx}{dt} = 2 k t$

• Total Distance (Integral): $X = \int_{t_1}^{t_N} v(t) dt$ For $v(t) = 2 k t$, $X = k t_N^2 - k t_1^2$

Table: Comparison of Speed and Velocity

Quantity	Definition	Type	Formula
Speed	Rate of change of distance	Scalar	$\bar{v} = \frac{l_{tot}}{t_{tot}}$
Velocity	Rate of change of displacement	Vector	$v = \frac{\Delta x}{\Delta t}$

Additional info:

• Some advanced calculus concepts (derivative and integral) are included for completeness, though not required for introductory physics.

• Examples and problems are based on real-world scenarios to illustrate the application of kinematic principles.