Motion in a Straight Line

Introduction

Motion in a straight line, also known as one-dimensional motion, is the simplest form of describing how an object changes its position over time. It forms the foundational concepts for understanding more complex motion in two or three dimensions. By analyzing motion along a straight line, we introduce fundamental kinematic quantities such as position, displacement, distance, velocity, and acceleration.

Understanding these quantities allows us to mathematically describe an object's movement, predict its future position, and analyze the causes of its motion (dynamics).

Position, Path Length And Displacement

To describe motion, we first need to define an object's location in space relative to a reference point. This is done using the concepts of position, path length, and displacement.

Path Length

Path length is the total distance covered by an object as it moves from its initial position to its final position. It is a scalar quantity, meaning it only has magnitude and no direction associated with it.

• Path length is always positive.
• It measures the total length of the trajectory traced by the object.
• Example: If a person walks 2 meters forward and then 1 meter backward, the total path length covered is $ 2 \, \text{m} + 1 \, \text{m} = 3 \, \text{m} $.

Displacement

Displacement is the change in the position of an object. It is a vector quantity, meaning it has both magnitude and direction. Displacement is defined as the final position minus the initial position.

• Mathematically, displacement $ \Delta x $ in one dimension is given by:

  $ \Delta x = x_f - x_i $

  where $ x_f $ is the final position and $ x_i $ is the initial position.
• Displacement indicates the net change in position, irrespective of the path taken.
• It can be positive, negative, or zero.
• Example: If the person in the previous example starts at $ x=0 $, moves to $ x=2 $, and then moves back to $ x=1 $, their initial position is $ x_i = 0 $ and their final position is $ x_f = 1 $. The displacement is $ \Delta x = x_f - x_i = 1 \, \text{m} - 0 \, \text{m} = 1 \, \text{m} $ (1 meter in the positive direction).

Key Distinction: While path length measures the total ground covered, displacement measures the net change in position from start to finish. For an object moving in a straight line without reversing direction, the magnitude of its displacement is equal to the path length. However, if the object changes direction, the path length will be greater than the magnitude of the displacement.

Average Velocity And Average Speed

Average velocity and average speed are used to describe the overall rate of motion over a period of time.

Average Velocity

Average velocity ($ \vec{v}_{avg} $) is defined as the displacement divided by the time interval during which the displacement occurred. It is a vector quantity.

• Formula:

  $ \vec{v}_{avg} = \frac{\Delta \vec{x}}{\Delta t} = \frac{\text{Displacement}}{\text{Time Interval}} $

  In one dimension, this is:

  $ v_{avg} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i} $

• Average velocity indicates the direction and average rate of change of position.
• If an object returns to its starting position, its displacement is zero, and therefore its average velocity is zero, even if it moved a considerable distance.

Average Speed

Average speed is defined as the total path length divided by the time interval during which the motion occurred. It is a scalar quantity.

• Formula:

  $ \text{Average Speed} = \frac{\text{Total Path Length}}{\text{Time Interval}} $

• Average speed tells us how fast the object was moving on average, regardless of its direction.
• Average speed is always non-negative.
• Relationship between Average Speed and Average Velocity: The magnitude of the average velocity is equal to the average speed only if the object moves in a straight line without reversing its direction. In general, the average speed is greater than or equal to the magnitude of the average velocity.

  $ \text{Average Speed} \ge |\vec{v}_{avg}| $

Example: A car travels 100 km east in 2 hours, then turns around and travels 50 km west in 1 hour.

• Total path length = 100 km + 50 km = 150 km.
• Total time interval = 2 hours + 1 hour = 3 hours.
• Average speed = $ \frac{150 \, \text{km}}{3 \, \text{h}} = 50 \, \text{km/h} $.
• Initial position (assuming origin is starting point) = 0. Final position = 100 km East - 50 km West = 50 km East.
• Displacement = 50 km East.
• Average velocity = $ \frac{50 \, \text{km East}}{3 \, \text{h}} \approx 16.7 \, \text{km/h East} $.
• Here, average speed (50 km/h) is greater than the magnitude of average velocity (16.7 km/h).

Instantaneous Velocity And Speed

While average velocity and speed describe the overall motion, instantaneous velocity and speed describe the motion at a particular moment in time.

Instantaneous Velocity

Instantaneous velocity ($ \vec{v} $) is the velocity of an object at a specific instant in time. It is the rate of change of position at that precise moment.

• It is found by taking the limit of the average velocity as the time interval approaches zero. In calculus terms, it is the derivative of the position with respect to time:

  $ \vec{v}(t) = \lim_{\Delta t \to 0} \frac{\Delta \vec{x}}{\Delta t} = \frac{d\vec{x}}{dt} $

• Instantaneous velocity is a vector quantity, having both magnitude (instantaneous speed) and direction at that moment.
• The direction of the instantaneous velocity is always tangent to the object's path at that point. For motion in a straight line, the direction is simply along that line (positive or negative).

Instantaneous Speed

Instantaneous speed is the magnitude of the instantaneous velocity. It tells us how fast an object is moving at a particular instant, without specifying direction.

• Formula:

  $ \text{Instantaneous Speed} = |\vec{v}(t)| = | \frac{d\vec{x}}{dt} | $

• Instantaneous speed is a scalar quantity.
• If an object is moving in a straight line without changing direction, its instantaneous speed is equal to the magnitude of its instantaneous velocity.

Example: When a car's speedometer reads 60 km/h, it is indicating the instantaneous speed. If the car is moving North, its instantaneous velocity is 60 km/h North.

Acceleration

Acceleration is the rate at which an object's velocity changes over time. Since velocity is a vector, acceleration can result from a change in speed, a change in direction, or both.

Average Acceleration

Average acceleration ($ \vec{a}_{avg} $) is defined as the change in velocity divided by the time interval over which the change occurred. It is a vector quantity.

• Formula:

  $ \vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_f - \vec{v}_i}{t_f - t_i} $

  where $ \vec{v}_f $ is the final velocity and $ \vec{v}_i $ is the initial velocity.
• Average acceleration indicates the overall rate of velocity change over a period.

Instantaneous Acceleration

Instantaneous acceleration ($ \vec{a} $) is the acceleration of an object at a specific instant in time. It is the rate of change of velocity at that precise moment.

• It is found by taking the limit of the average acceleration as the time interval approaches zero. In calculus terms, it is the derivative of the velocity with respect to time:

  $ \vec{a}(t) = \lim_{\Delta t \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d\vec{v}}{dt} $

• Instantaneous acceleration describes how the velocity is changing at any given moment.
• If the velocity is constant, the acceleration is zero.
• If the velocity is changing, there is acceleration. The direction of acceleration depends on how the velocity is changing (e.g., speeding up, slowing down, or changing direction).

Example: A car brakes from 100 km/h to 0 km/h. Its velocity is changing rapidly, meaning it has a large deceleration (negative acceleration). If a car turns a corner at a constant speed, its speed isn't changing, but its velocity is (because its direction is changing). This change in velocity implies an acceleration directed towards the center of the curve (centripetal acceleration).

Kinematic Equations For Uniformly Accelerated Motion

When an object moves in a straight line with constant acceleration, its motion can be described by a set of three kinematic equations. These equations relate displacement ($ s $), initial velocity ($ u $), final velocity ($ v $), acceleration ($ a $), and time ($ t $).

The three primary kinematic equations for motion with constant acceleration are:

1. Velocity-Time Relation:

   $ v = u + at $

   This equation relates final velocity to initial velocity, acceleration, and time.
2. Position-Time Relation:

   $ s = ut + \frac{1}{2}at^2 $

   This equation relates displacement to initial velocity, acceleration, and time.
3. Position-Velocity Relation:

   $ v^2 = u^2 + 2as $

   This equation relates final velocity to initial velocity, acceleration, and displacement, without involving time.

Where:

• $ s $ = displacement
• $ u $ = initial velocity
• $ v $ = final velocity
• $ a $ = constant acceleration
• $ t $ = time interval

Important Note: These equations are only valid for motion with constant acceleration.

Relative Velocity

Relative velocity is the velocity of an object as observed from a particular frame of reference. When we talk about the velocity of an object, it is always relative to something else (e.g., relative to the ground, relative to another moving object).

In one dimension:

Let $ v_{AB} $ be the velocity of object A relative to object B.

If object A is moving with velocity $ \vec{v}_A $ and object B is moving with velocity $ \vec{v}_B $, both measured with respect to a common reference frame (say, the ground, denoted by G), then the velocity of A relative to B is given by:

$ \vec{v}_{AB} = \vec{v}_A - \vec{v}_B $

Similarly, the velocity of B relative to A is:

$ \vec{v}_{BA} = \vec{v}_B - \vec{v}_A = - \vec{v}_{AB} $

Example:

Consider two cars, A and B, moving on a straight road.

• Let the ground be the reference frame.
• Car A is moving East at 60 km/h ($ \vec{v}_A = +60 \, \text{km/h} $).
• Car B is moving East at 40 km/h ($ \vec{v}_B = +40 \, \text{km/h} $).
• The velocity of car A relative to car B is:

  $ \vec{v}_{AB} = \vec{v}_A - \vec{v}_B = (+60) - (+40) = +20 \, \text{km/h} $

  So, from the perspective of someone in car B, car A appears to be moving East at 20 km/h.
• If car B was moving West at 40 km/h ($ \vec{v}_B = -40 \, \text{km/h} $):

  $ \vec{v}_{AB} = \vec{v}_A - \vec{v}_B = (+60) - (-40) = +100 \, \text{km/h} $

  In this case, car A appears to be moving East at 100 km/h relative to car B.

Relative velocity is a fundamental concept, especially when dealing with frames of reference that are themselves in motion.