Motion in One Dimension (Kinematics) – Study Notes

Motion in One Dimension (Kinematics)

Reference Frames

All measurements of position, distance, or speed must be made with respect to a reference frame. A reference frame provides a context for describing motion, including a coordinate system, an origin, a clock, and an observer.

• Coordinate system: Used to measure position, typically along an axis (e.g., x-axis).

• Origin: The starting point of measurement, often set to zero.

• Clock: Used to measure time intervals.

• Observer: The person or device making the measurement.

Example: If you are sitting on a train and someone walks down the aisle, their speed relative to the train is different from their speed relative to the ground.

Position

Position (x): The location of a particle with respect to a chosen reference point (such as the origin). Finding an object's position means determining its location relative to some reference, often the origin of a coordinate system.

• Position is measured in the SI unit of meters (m).

• Positions can be positive or negative depending on the chosen direction.

Example: If a car is at three different positions: $x_A = -30~m$, $x_B = 0~m$, $x_C = 20~m$

Distance and Displacement

When an object changes its position, it moves and has both distance and displacement.

• Distance (d): The total length of the path traveled, measured along the actual path.

• Displacement ($riangle x$): The change in position from the starting point to the ending point, regardless of the path taken.

• Both are measured in meters (m).

Formula for Displacement:

$riangle x = x_2 - x_1$

• $x_1$: initial position (m)

• $x_2$: final position (m)

• $riangle x$: displacement (m)

Example: An object moves 70 m east of the origin and returns 30 m west. Distance traveled: $d = 100~m$ Displacement: $riangle x = 40~m$

Direction: Motion to the right: Displacement is positive. Motion to the left: Displacement is negative.

Average Velocity

Speed (s): How far an object travels in a given time interval. Velocity (v): The speed in a specific direction, including directional information.

• Average speed: $s = rac{d}{t}$

• Average velocity: $v_{av} = rac{ riangle x}{ riangle t}$

• Speed is measured in meters per second (m/s) and does not require direction.

• Velocity is measured in meters per second (m/s) and must be specified with direction.

Example 1: Runner's average velocity: $x_1 = 50.0~m$, $x_2 = 30.5~m$, $riangle t = 3.00~s$ $riangle x = 30.5 - 50.0 = -19.5~m$ $v_{av} = rac{-19.5}{3} = -6.50~m/s$ Negative sign indicates motion to the left.

Example 2: Cyclist travels at 18 km/h for 2.5 h. $2.5~h imes 3600 = 9000~s$ $18~km/h imes rac{1000}{3600} = 5~m/s$ $riangle x = 5~m/s imes 9000~s = 45,000~m = 45 imes 10^3~m$

Example 3: Car travels 100 km at 50 km/h, then 100 km at 100 km/h. $riangle t_A = rac{100}{50} = 2~h$ $riangle t_B = rac{100}{100} = 1~h$ $s_{av} = rac{100 + 100}{2 + 1} = 66.7~km/h$

Table: Common Units in Kinematics

Quantity	Symbol	Unit
Distance/Displacement	d or $riangle x$	m
Time	t or $riangle t$	s
Speed/Velocity	s or v	m/s

Additional info:

• Reference frames are essential for describing motion accurately in physics.

• Displacement is a vector quantity, while distance is a scalar.

• Average velocity can be negative, zero, or positive depending on direction.

• Speed is always non-negative.