Kinematics in One Dimension: Motion Along a Straight Line

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 interval.

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

• 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: If a runner completes a 400 m lap in 50 s, average speed = 8 m/s; if the runner returns to the starting point, displacement = 0, so average velocity = 0.

Instantaneous Velocity

Instantaneous velocity describes the rate of change of position at a specific moment in time.

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

• Vector property: Velocity has both magnitude and direction.

• Significance: The sign of v indicates direction; positive for one direction, negative for the opposite.

• Graphical interpretation: Instantaneous velocity is the slope of the tangent to the x–t graph at a given time.

Instantaneous Acceleration

Acceleration measures how quickly velocity changes with time.

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

• Second derivative: Acceleration is the second derivative of position with respect to time.

• Example: For $x(t) = 5 + 4t + 3t^2$, $v(t) = 4 + 6t$, $a(t) = 6$ (constant acceleration).

Velocity and the Slope of the Tangent

• On a position vs. time graph, the slope of the tangent at any point gives the instantaneous velocity.

• Positive slope: Positive velocity (object moving forward).

• Negative slope: Negative velocity (object moving backward).

Types of Motion by Acceleration

• Instantaneous velocity can be:

  1. Constant during a time interval (e.g., cruise control).

  2. Continuously changing magnitude (speeding up or slowing down).

  3. Changing direction (since velocity is a vector).

• Cases 2 and 3 involve accelerated motion.

Equations of Motion: Constant Velocity

• For motion along a straight line with constant velocity:

• $v = \frac{x - x_0}{t}$

• $x = vt$

• Average velocity equals instantaneous velocity.

Equations of Motion: Constant Acceleration

• For motion with constant acceleration (uniformly accelerated motion):

• $a = \frac{v - v_0}{t}$

• $v = v_0 + at$

• $x = v_0 t + \frac{1}{2} a t^2$

• These equations allow calculation of velocity and position at any time, given initial conditions.

Galilei’s Formula (Third Equation)

• When time is not given or required, use:

• $v^2 = v_0^2 + 2a(x - x_0)$

• This relates velocity, acceleration, and displacement without involving time.

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

• Displacement is the area under the velocity vs. time graph.

• Change in velocity is the area under the acceleration vs. time graph.

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

• Example: For a velocity-time graph, the shaded area between t0 and t1 represents the displacement.

Free Fall: Example of Constant Acceleration

• All bodies near Earth's surface experience a downward acceleration of $g = 9.8\,\text{m/s}^2$ if air resistance is negligible.

• Direction is always downward, even if the body moves upward.

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

• Equations for free fall:

• $v = v_0 - gt$

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

• $v^2 = v_0^2 - 2g(y - y_0)$

• Example: A rock thrown upward from a bridge: use these equations to find its velocity and position at any time.

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 Acceleration: Given distance, time, and final speed, find initial speed and acceleration using kinematic equations.

• Block on Incline: Find the speed of a block at a certain position on a frictionless incline, given initial conditions and acceleration.

• Automobile and Train: Calculate time and distance for a car to overtake a train, using relative velocity concepts.

• Police Car and Speeder: Determine the time for a police car to catch up to a speeder, given acceleration and initial conditions.

Graphical Interpretation

• The area under the v–t graph gives displacement.

• The area under the a–t graph gives change in velocity.

• Negative areas indicate motion in the opposite direction.

Table: Summary of Kinematic Equations

Equation	Physical Meaning
$v = v_0 + at$	Velocity at time t
$x = v_0 t + \frac{1}{2} a t^2$	Position at time t
$v^2 = v_0^2 + 2a(x - x_0)$	Velocity-displacement relation (no time)
$x = \frac{1}{2}(v_0 + v)t$	Displacement using average velocity
$x = v t - \frac{1}{2} a t^2$	Displacement (missing initial velocity)

Additional info: These notes expand on the original slides and handwritten content, providing full definitions, equations, and context for each topic. All sample problems referenced are standard applications of the kinematic equations in one dimension.