Comprehensive Study Notes on Velocity and Acceleration

Definition and Characteristics of Velocity

Velocity is defined as a vector quantity, which distinguishes it significantly from speed. It is mathematically represented by the symbol $v$ or v\mathbf{v}. To fully specify the velocity of a moving body, both its magnitude (representing speed) and its direction of motion must be provided. Two bodies are considered to be moving with the same velocities only if they satisfy two conditions simultaneously: they must move with the same speed and they must move in the same direction. Conversely, if two bodies move with the same speed but in different directions, or if they move with different speeds in the same direction, they are characterized as moving with different velocities.

The units used to measure velocity are identical to those used for speed. In the International System of Units (S.I.), the unit of velocity is metre per second, denoted as $m\,s^{-1}$. In the Centimetre-Gram-Second (C.G.S.) system, the unit is centimetre per second, denoted as $cm\,s^{-1}$.

Uniform and Non-Uniform Velocity

A body is said to be moving with uniform velocity if it travels equal distances in a specific, unchanging direction within equal intervals of time. An example of this occurs when a body is set in motion on a frictionless surface; it will continue to move with a consistent velocity. Another significant instance is the fall of raindrops. Initially, as a rain drop begins to fall, its velocity increases due to the force of gravity. However, due to the viscosity (friction) and upthrust exerted by the air, a resistive force develops. This viscous force and upthrust eventually balance the force of gravity acting on the drop, resulting in a net force of zero. At this point, the drop continues to fall at a constant speed in a constant direction, known as the terminal velocity.

In cases where velocity is uniform, if a body completes a displacement $s$ within a time interval $t$, the velocity $v$ is defined by the following expression:

v=stv = \frac{s}{t}

From this definition, the displacement can be calculated as:

Displacement=v×t\text{Displacement} = v \times t

Non-uniform or variable velocity occurs when there is a change in the magnitude of velocity, its direction, or both. A body may cover unequal distances in a particular direction in equal time intervals, or it may cover equal distances in equal time intervals while its direction of motion changes. For example, a freely falling body exhibits variable velocity because, while its direction remains constant (downward), its speed continuously increases due to gravity. Similarly, a body moving in a circular path at a constant speed possesses variable velocity because its direction of motion is changing at every single point along the path.

Instantaneous and Average Velocity

For objects moving with non-uniform or variable velocity, motion is described using instantaneous and average values. Instantaneous velocity refers to the velocity of a body at any specific moment in time. It is determined by finding the ratio of the distance travelled within an extremely small time interval to that time interval itself. It is critical that the time interval chosen is small enough so that the direction of motion does not change during that period.

Average velocity is the measure used to describe the overall motion in a particular direction when velocity changes over time. It is defined as the ratio of the total displacement to the total time taken for the entire journey. The mathematical representation is:

Average velocity=DisplacementTotal time taken\text{Average velocity} = \frac{\text{Displacement}}{\text{Total time taken}}

Distinction Between Speed and Velocity

There are several fundamental differences between speed and velocity. First, speed is a scalar quantity, indicating only how fast a body is moving, whereas velocity is a vector quantity that describes both the rate of motion and the direction. Second, for motion along a straight line, the magnitude of velocity is equal to the speed. While speed is always a positive value, velocity is assigned a positive or negative sign depending on the chosen direction of motion.

Another critical distinction lies in their average values over a journey. The average velocity of a body can be zero even if its average speed is non-zero. For instance, if a body starts its motion from a specific point and returns to that same point after a certain time, its displacement is zero. Consequently, the average velocity is zero (0/t=00/t = 0), even though the total distance travelled and the average speed are non-zero.

Dynamics of Circular Motion

Circular motion is a specific case of two-dimensional or non-linear motion. Even when a body moves along a circular path with a uniform speed, it is categorized as having variable velocity. This is because the direction of motion changes continuously at a uniform rate as the body traverses the circle. At any given instant, the direction of the velocity is along the tangent to the circular path at that specific point.

In a circular path of radius $r$, if a body covers equal distances in equal time intervals, the speed is considered uniform. However, after completing one full round in a time $T$, the displacement is zero because the starting and ending points are identical. Thus, the average velocity for one complete round is zero. In contrast, the average speed for that same round is calculated based on the circumference of the path:

Average speed=2πrT\text{Average speed} = \frac{2\pi r}{T}

Acceleration and Retardation

Acceleration is defined as the rate of change of velocity with respect to time. Numerically, it represents the change in velocity that occurs in one second. The formula for acceleration is:

Acceleration=Change in velocityTime interval\text{Acceleration} = \frac{\text{Change in velocity}}{\text{Time interval}}

The S.I. unit for acceleration is derived by dividing the unit of velocity ($m\,s^{-1}$) by the unit of time ($s$), resulting in metre per second squared, or $m\,s^{-2}$. In the C.G.S. system, the unit is $cm\,s^{-2}$.

Most objects do not move with uniform velocities; their motion is typically variable, such as a car moving through a busy market or a satellite in orbit. If we consider motion strictly in a straight line (where direction does not change), any change in velocity is due to a change in speed. If the velocity increases over time, the motion is "accelerated." If the velocity decreases over time, the motion is "decelerated" or "retarded."

Retardation is essentially negative acceleration. While acceleration signifies an increase in velocity per second, retardation indicates a decrease in velocity per second.

Mathematical Relations and Numerical Examples

To derive the mathematical relationship for acceleration, let a body move in a straight line with an initial velocity $u$. After a time interval $t$, its final velocity becomes $v$. The change in velocity is represented by the expression $(v - u)$. Therefore, acceleration $a$ is:

a=v−uta = \frac{v - u}{t}

This equation can be rearranged to find the final velocity:

v=u+atv = u + at

In this context, if the final velocity $v$ is greater than the initial velocity $u$ (v>uv > u), then the acceleration $a$ is positive. If the final velocity is less than the initial velocity (v<uv < u), then $a$ is negative, which represents retardation.

Consider the following numerical example: A car starts from rest (initial velocity u=0 m s−1u = 0\,m\,s^{-1}) and acquires a velocity of 20 m s−120\,m\,s^{-1} in a time of 10 s10\,s. The change in velocity is:

Change in velocity=20 m s−1−0 m s−1=20 m s−1\text{Change in velocity} = 20\,m\,s^{-1} - 0\,m\,s^{-1} = 20\,m\,s^{-1}

The acceleration of the car is then calculated as:

a=20 m s−1−0 m s−110 sa = \frac{20\,m\,s^{-1} - 0\,m\,s^{-1}}{10\,s}

a=20 m s−110 sa = \frac{20\,m\,s^{-1}}{10\,s}

a=2 m s−2a = 2\,m\,s^{-2}