Kinematics in One Dimension: Structured Study Notes

Chapter 2: Describing Motion – Kinematics in One Dimension

2.1 Reference Frames and Displacement

Kinematics is the study of how objects move, focusing on position, velocity, and acceleration. All measurements of position, distance, or speed must be made with respect to a reference frame, which is a coordinate system used to define where and how motion is observed.

• Reference Frame: The perspective from which motion is measured (e.g., ground, moving train).

• Displacement: The change in position of an object from its starting point to its ending point. Displacement is a vector (has magnitude and direction).

• Distance: The total length of the path traveled, regardless of direction. Distance is a scalar (has magnitude only).

• Formula for Displacement: $\Delta x = x_2 - x_1$

• Sign Convention:

  • Displacement is positive if moving in the positive direction of the chosen axis.

  • Displacement is negative if moving in the negative direction.

• Example: If a person moves from $x_1 = 10$ m to $x_2 = 30$ m, $\Delta x = 20$ m (positive). If moving from $x_1 = 30$ m to $x_2 = 10$ m, $\Delta x = -20$ m (negative).

2.2 Average Velocity

Average velocity describes how fast and in what direction an object's position changes over a time interval. Speed is the magnitude of velocity and does not include direction.

• Average Speed Formula: $\text{average speed} = \frac{\text{distance traveled}}{\text{time elapsed}} = \frac{L}{\Delta t}$

• Average Velocity Formula: $\text{average velocity} = \frac{\text{displacement}}{\text{time elapsed}} = \frac{x_f - x_i}{\Delta t}$

• Example: A runner moves from $x_1 = 50.0$ m to $x_2 = 30.5$ m in $3.00$ s. $\text{average velocity} = \frac{30.5 - 50.0}{3.00} = -6.5$ m/s

2.3 Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific instant in time. It is the limit of average velocity as the time interval approaches zero.

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

• Graphical Interpretation: On a position vs. time graph, instantaneous velocity is the slope of the tangent to the curve at a point.

• Example: For $x(t) = 5t^2 + 3t - 1$, $v(t) = \frac{dx}{dt} = 10t + 3$

• Instantaneous Speed: Always equals the magnitude of instantaneous velocity.

2.4 Acceleration

Acceleration is the rate at which velocity changes with time. It can be average or instantaneous.

• Average Acceleration Formula: $\text{average acceleration} = \frac{v_2 - v_1}{t_2 - t_1}$

• Instantaneous Acceleration Formula: $a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$

• Negative Acceleration vs. Deceleration:

  • Negative acceleration: Acceleration in the negative direction of the coordinate system.

  • Deceleration: Acceleration opposite to the direction of velocity.

• Example: For $x(t) = 5t^2 + 3t - 1$, $v(t) = 10t + 3$ $a(t) = \frac{dv}{dt} = 10$ (constant acceleration)

2.5 Motion at Constant Acceleration

When acceleration is constant, the motion of an object can be described by a set of kinematic equations.

• Average Velocity: $\bar{v} = \frac{\Delta x}{\Delta t} = \frac{x - x_0}{t}$

• Constant Acceleration: $a = \frac{v - v_0}{t}$

• Velocity as a Function of Time: $v = v_0 + at$

• Position as a Function of Time: $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• Velocity-Position Relation: $v^2 = v_0^2 + 2a(x - x_0)$

• Average Velocity (for constant acceleration): $\bar{v} = \frac{v_0 + v}{2}$

• Sign Convention: The values of $x_0$, $v_0$, and $a$ can be positive or negative depending on the chosen coordinate system.

2.6 Solving Problems

Solving kinematics problems involves systematic steps to ensure accuracy and understanding.

1. Read the entire problem carefully.

2. Decide on the objects under study and the relevant time interval.

3. Draw a diagram and choose coordinate axes.

4. Write down known (given) quantities and identify unknowns.

5. Plan an approach using relevant physics principles.

6. Identify equations relating knowns and unknowns; check their validity.

7. Calculate the solution, rounding appropriately.

8. Evaluate the result for reasonableness and consistency with estimates.

9. Check units for correctness.

Example: Acceleration of a Car

A car accelerates from rest ($v_0 = 0$) at $a = 2.00$ m/s2 to cross a $30.0$ m intersection. Find the time required.

2.7 Freely Falling Objects

Objects near Earth's surface experience a constant acceleration due to gravity, denoted as $g$.

• Acceleration due to Gravity: $g \approx 9.80$ m/s2 (downward)

• Free Fall: In the absence of air resistance, all objects fall with the same acceleration, regardless of mass.

• Effects of Air Resistance: Air resistance can cause lighter objects or those with larger surface area to fall more slowly.

• Example:

  • Ball thrown downward with initial velocity $v_0 = 3.00$ m/s: Find position and speed after $1.00$ s and $2.00$ s.

  • Ball thrown upward with $v_0 = 15.0$ m/s: Calculate maximum height and total time in air (ignoring air resistance).

Summary Table: Kinematic Quantities

Quantity	Definition	Formula	Type
Displacement	Change in position	$\Delta x = x_2 - x_1$	Vector
Distance	Length of path traveled	Sum of all path segments	Scalar
Average Velocity	Displacement per time	$\bar{v} = \frac{\Delta x}{\Delta t}$	Vector
Average Speed	Distance per time	$\text{average speed} = \frac{L}{\Delta t}$	Scalar
Instantaneous Velocity	Velocity at a moment	$v = \frac{dx}{dt}$	Vector
Average Acceleration	Change in velocity per time	$a = \frac{v_2 - v_1}{t_2 - t_1}$	Vector
Instantaneous Acceleration	Acceleration at a moment	$a = \frac{dv}{dt}$	Vector
Gravity (free fall)	Acceleration due to gravity	$g = 9.80$ m/s2	Vector (downward)

Summary of Key Concepts

• Kinematics describes motion using position, velocity, and acceleration.

• Reference frames are essential for defining motion.

• Displacement and distance differ in directionality and physical meaning.

• Velocity and speed are distinct; velocity includes direction.

• Acceleration quantifies changes in velocity.

• Equations of motion for constant acceleration allow prediction of future position and velocity.

• All objects near Earth's surface fall with the same acceleration in the absence of air resistance.

Additional info: These notes expand on the provided slides and text, adding definitions, formulas, and examples for clarity and completeness.