Motion in One Dimension: Kinematics and Applications

Motion in One Dimension

Basic Quantities in Mechanics

Mechanics begins with the study of motion, using several fundamental quantities:

• Mass (kg): A measure of the amount of matter in an object.

• Length (m): The spatial distance between two points.

• Time (s): The ongoing sequence of events taking place.

Derived quantities are built from these, such as:

• Velocity (m/s): The rate of change of position with respect to time.

• Acceleration (m/s2): The rate of change of velocity with respect to time.

Note: The position of an object is often described using a coordinate system (e.g., x-axis for one-dimensional motion).

Displacement

Displacement is the change in position of an object:

• It is a vector quantity (has both magnitude and direction).

• Defined as: $\Delta x = x_{\text{final}} - x_{\text{initial}}$

• Units: meters (m)

Example: If a car moves from $x_1 = 20\,\text{m}$ to $x_2 = 50\,\text{m}$, the displacement is $\Delta x = 50 - 20 = 30\,\text{m}$ to the right.

Distance vs. Displacement

• Distance is the total length of the path traveled, regardless of direction (scalar).

• Displacement is the straight-line change in position (vector).

Example: If a car moves 10 m right, then 5 m left, the distance is 15 m, but the displacement is 5 m right.

Average Velocity and Speed

• Average velocity is the displacement divided by the time interval:

$v_{\text{avg}} = \frac{\Delta x}{\Delta t}$

• Units: m/s

• Direction matters (sign indicates direction).

• Average speed is the total distance traveled divided by the time interval (scalar):

$\text{Average speed} = \frac{\text{Total distance}}{\Delta t}$

Instantaneous Velocity

• The velocity at a specific instant in time.

• Defined as the derivative of position with respect to time:

$v = \frac{dx}{dt}$

Graphical Interpretation

• The slope of the $x$ vs. $t$ graph at any point gives the instantaneous velocity.

• The area under the $v$ vs. $t$ graph gives the displacement.

Acceleration

Acceleration is the rate of change of velocity with respect to time:

• Average acceleration:

$a_{\text{avg}} = \frac{\Delta v}{\Delta t}$

• Instantaneous acceleration:

$a = \frac{dv}{dt}$

• Units: m/s2

• Can be positive or negative (sign indicates direction).

Equations of Motion for Constant Acceleration

For motion with constant acceleration $a$, the following equations apply:

• $v = v_0 + at$

• $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• $v^2 = v_0^2 + 2a(x - x_0)$

Where:

• $x_0$ = initial position

• $v_0$ = initial velocity

• $a$ = constant acceleration

• $t$ = time elapsed

Special Case: Free Fall

• In the absence of air resistance, all objects near Earth's surface fall with the same constant acceleration $g \approx 9.80\,\text{m/s}^2$ downward.

• Equations of motion for free fall (taking upward as positive):

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

$v = v_0 - g t$

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

Worked Examples

• Example 1: Calculating average velocity for a car trip with stops and varying speeds.

• Example 2: Determining stopping distance for a car decelerating at a constant rate.

• Example 3: Finding the time and height for a ball thrown upward, using kinematic equations.

• Example 4: Calculating the drop distance for a stone released from rest from a building.

Summary Table: Kinematic Equations for Constant Acceleration

Equation	Physical Meaning
$v = v_0 + at$	Velocity after time $t$
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	Position after time $t$
$v^2 = v_0^2 + 2a(x - x_0)$	Relates velocity and displacement (no time)

Key Points to Remember

• Displacement and velocity are vectors; direction matters.

• Speed and distance are scalars; only magnitude matters.

• For constant acceleration, use the kinematic equations to solve for unknowns.

• In free fall, acceleration is always $-g$ (downward).

Additional info:

• When solving problems, always define the coordinate system and assign positive/negative directions.

• Check units for consistency in all calculations.