Units and Measurements

2.1 Introduction

Motion is a universal phenomenon observed in all objects and living beings. From humans walking or running, to natural processes like leaves falling and water flowing, motion is everywhere. Even when we are asleep, air moves in and out of our lungs and blood flows through arteries and veins. Vehicles like automobiles and airplanes transport people from one place to another. On a cosmic scale, the Earth rotates on its axis every 24 hours and revolves around the Sun once a year. The Sun itself moves within the Milky Way galaxy, which is also moving within a group of galaxies. Motion is defined as the change in position of an object with respect to time. To describe motion scientifically, we need to understand how position changes with time. This chapter focuses on describing motion along a straight line, known as rectilinear motion. We will develop concepts such as velocity and acceleration to describe motion quantitatively. For rectilinear motion with uniform acceleration, simple equations can be derived to relate displacement, velocity, acceleration, and time. Finally, the concept of relative velocity will be introduced to understand the relative nature of motion. In our study, objects in motion will be treated as point objects. This means we consider the object's size negligible compared to the distance it moves in the given time. This approximation is valid in many real-life situations and simplifies analysis without significant error. Kinematics is the branch of physics that deals with describing motion without considering its causes. The causes of motion, such as forces, are studied in later chapters (Chapter 4 onwards).

• Motion is the change in position of an object with time.
• Rectilinear motion refers to motion along a straight line.
• Objects are approximated as point objects when their size is negligible compared to the distance moved.
• Kinematics describes motion without addressing its causes.
• The Earth’s rotation and revolution are examples of motion on a cosmic scale.
• Relative velocity explains motion from different frames of reference.
• 📌 Motion: Change in position of an object with time.
• 📌 Rectilinear motion: Motion along a straight line.
• 📌 Point object: An object whose size is negligible compared to the distance moved.

2.2 Instantaneous velocity and speed

Average velocity over a time interval gives the overall rate of change of position but does not describe how velocity varies at each instant within that interval. To capture this, instantaneous velocity is defined as the velocity of an object at a specific instant of time. Mathematically, instantaneous velocity v at time t is defined as the limit of the average velocity as the time interval Δt approaches zero: v = lim (Δt → 0) (Δx / Δt) = dx/dt Here, dx/dt is the derivative of position x with respect to time t, representing the rate of change of position at that instant. This definition uses calculus to find the exact velocity at any moment. Graphically, instantaneous velocity at time t can be found as the slope of the tangent to the position-time graph at that point. For example, in Fig. 2.1, the velocity at t = 4 s is the slope of the tangent drawn at that instant on the position-time curve. Numerically, one can calculate average velocities over smaller and smaller intervals centered at t = 4 s, as shown in Table 2.1. As Δt decreases from 2 s to 0.01 s, the average velocity approaches a limiting value of approximately 3.84 m/s, which is the instantaneous velocity at t = 4 s. If the position as a function of time is known explicitly, differentiation can be used directly to find instantaneous velocity. For example, if x = a + b t², then v = dx/dt = 2 b t. Instantaneous speed is the magnitude of instantaneous velocity. For example, velocities of +24 m/s and -24 m/s both correspond to a speed of 24 m/s. While average speed over a finite interval is always greater than or equal to the magnitude of average velocity, instantaneous speed equals the magnitude of instantaneous velocity at any instant. **Table on page 2 (6×7)** | $\Delta t$ (s) | $t_1$ (s) | $t_2$ (s) | $x(t_1)$ (m) | $x(t_2)$ (m) | $\Delta x$ (m) | $\Delta x / \Delta t$ (m s$^{-1}$) | | --- | --- | --- | --- | --- | --- | --- | | 2.0 | 3.0 | 5.0 | 2.16 | 10.0 | 7.84 | 3.92 | | 1.0 | 3.5 | 4.5 | 3.43 | 7.29 | 3.86 | 3.86 | | 0.5 | 3.75 | 4.25 | 4.21875 | 6.14125 | 1.9225 | 3.845 | | 0.1 | 3.95 | 4.05 | 4.93039 | 5.31441 | 0.38402 | 3.8402 | | 0.01 | 3.995 | 4.005 | 5.100824 | 5.139224 | 0.0384 | 3.8400 | **Table on page 10 (3×5)** | Physical quantity | Symbol | Dimensions | Unit | Remarks | | --- | --- | --- | --- | --- | | Path length | | [L] | m | | | Displacement | Δx | [L] | m | = x_{0} - x_{1} **Table on page 10 (2×5)** | Velocity | | [LT^{4}] | m s^{-1} | | | (a) Average | v | | | = \frac{\Delta x}{\Delta t} | | (b) Instantaneous | v | | | = \frac{\lim}{\Delta t} \cdot \frac{\Delta x}{\Delta t} = \frac{\mathrm{d}x}{\mathrm{d}t} **Table on page 11 (1×5)** | --- | --- | --- | --- | --- | | Acceleration

• Instantaneous velocity is the velocity at a specific instant of time.
• It is defined as the limit of average velocity as the time interval approaches zero.
• Mathematically, v = dx/dt, the derivative of position with respect to time.
• Graphically, instantaneous velocity is the slope of the tangent to the position-time graph.
• Instantaneous speed is the magnitude of instantaneous velocity.
• Average speed over an interval is ≥ magnitude of average velocity; instantaneous speed equals magnitude of instantaneous velocity.
• 📌 Instantaneous velocity: Velocity of an object at a particular instant.
• 📌 Average velocity: Displacement divided by time interval.
• 📌 Speed: Magnitude of velocity.