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Kinematics in One Dimension: Motion Along a Straight Line

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Chapter 2 – Kinematics in One Dimension

Overview

Kinematics in one dimension studies the motion of objects along a straight line, focusing on position, displacement, velocity, acceleration, and the mathematical relationships between these quantities. This chapter introduces the foundational concepts and equations necessary to analyze linear motion, including graphical interpretations and applications to real-world scenarios such as free fall.

Position and Displacement

  • Position (x): The location of an object along a straight line, typically measured from a chosen origin.

  • Displacement (Δx): The change in position of an object; a vector quantity defined as final position minus initial position.

  • Example: If a car moves from x = 2 m to x = 7 m, its displacement is Δx = 7 m – 2 m = 5 m.

Average Velocity and Speed

  • Average velocity (vave): The total displacement divided by the total time taken.

  • Formula: $v_{ave} = \frac{\Delta x}{\Delta t}$

  • Speed: The magnitude of velocity; a scalar quantity representing how fast an object moves regardless of direction.

  • Distance travelled: The total length of the path taken, which may differ from displacement if the path is not straight.

  • Example: In a race, the turtle’s average speed is 0.9 m/s and the rabbit’s is 9 m/s over a distance of 1500 m.

Instantaneous Velocity

  • Definition: The velocity of an object at a specific instant.

  • Formula: $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$

  • Properties:

    • Vector quantity: Has both magnitude and direction.

    • The sign of v indicates direction.

    • Instantaneous speed is the magnitude of instantaneous velocity.

    • On an x-t graph, v is the slope of the tangent at a given point.

  • Example: If x(t) = 5 + 4t + 3t2, then v(t) = 4 + 6t.

Instantaneous Acceleration

  • Definition: The rate of change of velocity at a specific instant.

  • Formula: $a = \frac{dv}{dt} = \frac{d}{dt} \left( \frac{dx}{dt} \right ) = \frac{d^2x}{dt^2}$

  • Second derivative of position with respect to time.

  • Example: For x(t) = 5 + 4t + 3t2, a(t) = 6 m/s2.

Graphical Interpretation of Motion

  • Velocity and slope: The instantaneous velocity at a given time is the slope of the tangent to the x-t curve at that time.

  • Positive slope: Positive velocity; Negative slope: Negative velocity.

  • Displacement: The area under the v-t graph between two times represents the change in position.

  • Change in velocity: The area under the a-t graph between two times represents the change in velocity.

Types of Motion by Acceleration

  • Constant velocity: No acceleration; velocity remains unchanged.

  • Changing magnitude: Acceleration present; speed increases or decreases.

  • Changing direction: Acceleration present; direction of velocity changes.

  • Cases 2 and 3: Both involve accelerated motion.

Equations of Motion

Case 1: Constant Velocity

  • Definition: Motion along a straight line with constant direction and magnitude.

  • Equation: $v = \frac{x - x_0}{t}$

  • Position as a function of time: $x = vt$

  • Average velocity equals instantaneous velocity: $\vec{v}_{ave} = \vec{v}$

Case 2: Constant Acceleration (Uniformly Accelerated Motion)

  • Definition: Acceleration remains constant throughout the motion.

  • Equation for velocity: $v = v_0 + at$

  • Equation for position: $x = v_0 t + \frac{1}{2} a t^2$

  • Third equation (Galilei’s formula): $v^2 = v_0^2 + 2a(x - x_0)$

  • These equations can be used to solve any problem involving constant acceleration.

The Five Kinematic Equations

Equation

Missing Variable

$v = v_0 + at$

x

$x = v_0 t + \frac{1}{2} a t^2$

v

$v^2 = v_0^2 + 2a x$

t

$x = \frac{1}{2}(v_0 + v)t$

a

$x = v t - \frac{1}{2} a t^2$

$v_0$

Graphical Representation of Motion

  • Average velocity: $v_{average} = \frac{1}{2}(v + v_0)$

  • Displacement: Area under the velocity-time graph.

  • Change in velocity: Area under the acceleration-time graph.

  • If the area is below the axis, the value is negative.

Free Fall: Example of Motion with Constant Acceleration

  • Definition: Motion of objects under the influence of gravity alone, near Earth's surface.

  • Acceleration due to gravity: $a = 9.8\, \text{m/s}^2 = g$ (downwards)

  • If positive direction is upwards, $a = -g$ in all equations.

  • Equations for free fall:

    • $v = v_0 + at = v_0 - gt$

    • $y = y_0 + v_0 t + \frac{1}{2} a t^2 = y_0 + v_0 t - \frac{1}{2} g t^2$

    • $v^2 = v_0^2 + 2a(y - y_0) = v_0^2 - 2g(y - y_0)$

  • Example: A rock is thrown upward from a bridge; its velocity and position at various times can be found using these equations.

Sample Problems

  • Turtle and Rabbit Race: Calculate the maximum time the rabbit can wait before starting to run and still win, given their speeds and race distance.

  • Antelope with Constant Acceleration: Find the speed at the first point and the acceleration, given distance, time, and final speed.

  • Block on Frictionless Incline: Find the speed at a given position, knowing initial rest, distance traveled, and final speed.

  • Automobile and Train: Calculate time and distance for a car to overtake a train, given their speeds and the train's length.

  • Police Car and Speeder: Determine the time for a police car to catch up to a speeder, given their speeds and the police car's acceleration.

Additional info:

  • Graphical methods are essential for visualizing motion and interpreting kinematic quantities.

  • All kinematic equations assume constant acceleration unless otherwise specified.

  • Direction conventions (positive/negative) must be chosen and applied consistently in calculations.

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