Motion Along a Straight Line: Kinematics in One Dimension

Chapter 2: Motion Along a Straight Line

Introduction to Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. In this chapter, we focus on motion along a straight line (one-dimensional motion), introducing key concepts such as displacement, velocity, and acceleration.

• Kinematics: The study of how objects move.

• Velocity and acceleration are fundamental physical quantities in describing motion.

• Example: A sprinter accelerates at the start of a race and decelerates after crossing the finish line.

Displacement, Time, and Average Velocity

Definitions and Formulas

• Displacement ($\Delta x$): The change in position of a particle along the x-axis.

• Formula: $\Delta x = x_2 - x_1$

• Time Interval ($\Delta t$): The difference between the final and initial times.

• Formula: $\Delta t = t_2 - t_1$

• Average Velocity ($v_{\text{av},x}$): The displacement divided by the time interval.

• Formula: $v_{\text{av},x} = \frac{\Delta x}{\Delta t}$

Example: If a car moves from $x_1 = 10\,\text{m}$ to $x_2 = 50\,\text{m}$ in $t_2 - t_1 = 2\,\text{s}$, then $v_{\text{av},x} = \frac{50 - 10}{2} = 20\,\text{m/s}$.

Rules for the Sign of X-Velocity

If x-coordinate is:	... x-velocity is:
Positive and increasing (getting more positive)	Positive
Positive and decreasing (getting less positive)	Negative
Negative and increasing (getting less negative)	Positive
Negative and decreasing (getting more negative)	Negative

Position-Time (x-t) Graphs

Understanding x-t Graphs

• The slope of the x-t graph at any point gives the velocity at that instant.

• A straight line indicates constant velocity; a curved line indicates changing velocity.

• Slope zero: $v_x = 0$ (object at rest).

• Slope positive: $v_x > 0$ (object moving in positive x-direction).

• Slope negative: $v_x < 0$ (object moving in negative x-direction).

Instantaneous Velocity

Definition and Calculation

• Instantaneous velocity is the velocity at a specific instant or point along the path.

• Formula: $v_x = \frac{dx}{dt}$

• It is the slope of the tangent to the x-t curve at a given point.

• Average speed is not necessarily the magnitude of average velocity.

Example: If $x(t) = 3t^2$, then $v_x = \frac{d}{dt}(3t^2) = 6t$.

Average and Instantaneous Acceleration

Average Acceleration

• Acceleration describes the rate of change of velocity with time.

• Average x-acceleration: $a_{\text{av},x} = \frac{\Delta v_x}{\Delta t}$

Instantaneous Acceleration

• Instantaneous acceleration: $a_x = \frac{dv_x}{dt}$

• It is the slope of the tangent to the $v_x$-t curve at a given point.

Example: If $v_x(t) = 4t$, then $a_x = \frac{d}{dt}(4t) = 4$ (constant acceleration).

Rules for the Sign of X-Acceleration

If x-velocity is:	... x-acceleration is:
Positive and increasing (getting more positive)	Positive
Positive and decreasing (getting less positive)	Negative
Negative and increasing (getting less negative)	Positive
Negative and decreasing (getting more negative)	Negative

Motion with Constant Acceleration

Key Concepts

• If acceleration $a_x$ is constant, the velocity changes by equal amounts in equal time intervals.

• The position changes by different amounts in equal time intervals because velocity is changing.

• The area under the $a_x$-t graph gives the change in velocity.

Equations of Motion with Constant Acceleration

The following equations apply to straight-line motion with constant acceleration $a_x$:

Equation	Includes Quantities
$v_x = v_{0x} + a_x t$	$t, v_x, v_{0x}, a_x$
$x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$	$t, x, x_0, v_{0x}, a_x$
$v_x^2 = v_{0x}^2 + 2a_x(x - x_0)$	$x, x_0, v_x, v_{0x}, a_x$
$x - x_0 = \frac{1}{2}(v_{0x} + v_x)t$	$t, x, x_0, v_x, v_{0x}$

Example: A car starts from rest ($v_{0x} = 0$) and accelerates at $2\,\text{m/s}^2$ for $5\,\text{s}$. Its final velocity is $v_x = 0 + 2 \times 5 = 10\,\text{m/s}$.

Freely Falling Objects

Free Fall

• Free fall is the motion of an object under the influence of gravity alone (neglecting air resistance).

• All objects in free fall near Earth's surface experience the same constant acceleration downward, $g \approx 9.8\,\text{m/s}^2$.

• The velocity changes by the same amount in each time interval.

Equations for Free Fall

• Downward acceleration: $a_y = -g$ (if upward is positive direction).

• For an object dropped from rest: $y = y_0 - \frac{1}{2}gt^2$

• For an object thrown upward: $v_y = v_{0y} - gt$

Example: A coin dropped from a height falls with $a_y = -9.8\,\text{m/s}^2$ (ignoring air resistance).

Up-and-Down Motion in Free Fall

• When an object is thrown upward, it slows down under gravity, stops momentarily at the highest point, and then accelerates downward.

• The vertical velocity is zero at the highest point, but the acceleration is always $-g$.

Example: A ball thrown upward with $v_{0y} = 15.0\,\text{m/s}$ reaches its highest point when $v_y = 0$.

Additional info: The equations and concepts here are foundational for all further study in mechanics, including projectile motion and dynamics.