Chapter 2: Motion Along a Straight Line – Study Notes

Motion Along a Straight Line

Introduction to Motion

Motion is a fundamental concept in physics, describing how objects change their position over time. In this chapter, we focus on motion along a straight line, also known as one-dimensional motion. The study of motion can be divided into two main areas: kinematics and dynamics.

• Kinematics: Describes the movement of objects without considering the causes of motion. This is the primary focus of this chapter.

• Dynamics: Explains why objects move, which will be covered in later chapters.

Key Concepts and Goals

• Understand and use the concepts of displacement, velocity, and acceleration.

• Explore motions with constant acceleration.

• Interpret and construct graphs that describe motion.

• Examine the special case of freely falling bodies.

Vectors and Scalars

Definitions

• Vector Quantities: Have both magnitude and direction.

• Scalar Quantities: Have only magnitude, with no direction.

Examples of Vectors

• Traveling 72 mph due South

• Traveling 62 mph due North

• Lifting 50 lbs upward

• Pushing or pulling with 10 lbs of force in a specific direction

• Applying torque (e.g., 25 ft-lbs clockwise)

Examples of Scalars

• Traveling 72 mph (no direction specified)

• Applying 50 lbs (no direction specified)

• Applying 10 lbs (no direction specified)

• Torque magnitude (e.g., 25 ft-lbs, direction not specified)

Sign Convention for Vectors

In physics, positive and negative signs indicate direction, not quantity. For example:

• Car A: -70 mph (moving in the negative direction, e.g., west or south)

• Car B: +45 mph (moving in the positive direction, e.g., east or north)

The sign helps distinguish the direction of motion along a chosen axis.

Describing Motion

What is Motion?

Motion is the change in an object's position over time. In mechanics, motion is described using three key quantities:

• Displacement

• Velocity

• Acceleration

Displacement and Distance

• Displacement: The change in position of an object. It is a vector quantity, meaning it has both magnitude and direction.

• Distance: The total length of the path traveled by the object, regardless of direction. It is a scalar quantity.

Mathematical Expression for Displacement:

• For two points: $\Delta d = d_2 - d_1$

• For multiple segments: $\Delta d_{net} = \Delta d_1 + \Delta d_2 + \Delta d_3 + \ldots$

Mathematical Expression for Distance:

• $\text{Distance} = |\Delta d_1| + |\Delta d_2| + |\Delta d_3| + \ldots$

Units: Both displacement and distance are measured in meters (m) in the SI system.

Example: Displacement vs. Distance

If a person walks from their hometown to a university via a winding path, the distance is the total path length, while the displacement is the straight-line distance from the starting point to the final location.

Velocity and Speed

• Velocity: The rate of change of displacement with respect to time. It is a vector quantity.

• Speed: The rate at which distance is covered. It is a scalar quantity and always positive.

Average Velocity:

• $v_{ave} = \frac{x_2 - x_1}{t_2 - t_1}$

Instantaneous Velocity:

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

Average Speed:

• $\text{speed} = \frac{\text{distance}}{\Delta t}$

Units: Both velocity and speed are measured in meters per second (m/s).

Example: Calculating Average Velocity

If you and a friend run a 40-km race, and your friend runs at a steady 5.8 m/s, you can calculate the average velocity needed to finish 10 minutes earlier by solving for your required speed.

Acceleration

• Acceleration: The rate of change of velocity with respect to time. It is a vector quantity.

Average Acceleration:

• $a_{ave} = \frac{v_2 - v_1}{t_2 - t_1}$

Instantaneous Acceleration:

• $a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t}$

Units: Acceleration is measured in meters per second squared (m/s2).

Interpretation of Acceleration

• Acceleration can result in speeding up, slowing down, or changing direction.

• If acceleration and velocity are in the same direction, speed increases.

• If they are in opposite directions, speed decreases.

Motion with Constant Acceleration

Many physical situations involve constant acceleration, such as free fall. In these cases, the following kinematic equations apply:

• $v = v_0 + a t$

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

• $v^2 = v_0^2 + 2a(x - x_0)$

• $\text{Average velocity: } v_{avg} = \frac{v_0 + v}{2}$

Example: Reaction Time and Stopping Distance

If a car traveling at 24.5 m/s has a driver's reaction time of 0.70 s and decelerates at 3.75 m/s2 after braking, you can calculate:

• Distance covered during reaction time: $d_{reaction} = v_{initial} \times t_{reaction}$

• Distance covered during braking: $d_{brake} = \frac{v_{initial}^2}{2a}$ (with a negative acceleration)

• Total stopping distance: $d_{total} = d_{reaction} + d_{brake}$

Free Fall

Objects in free fall experience constant acceleration due to gravity, regardless of their mass (ignoring air resistance). On Earth, the acceleration due to gravity is:

• $g = -9.81 \text{ m/s}^2$ (downward)

The kinematic equations for free fall are the same as for constant acceleration, with $a = g$.

Example: Free Fall Calculations

• Time to reach maximum height when thrown upward: $t = \frac{v_0}{|g|}$

• Maximum height: $h_{max} = \frac{v_0^2}{2|g|}$

• Velocity at a given height: Use $v^2 = v_0^2 + 2g(y - y_0)$

Graphical Representation of Motion

Position-Time Graphs

• The slope of a position-time graph gives the velocity.

• A straight line indicates constant velocity.

• A curved line indicates changing velocity (acceleration).

Velocity-Time Graphs

• The slope of a velocity-time graph gives the acceleration.

• The area under the curve gives the displacement.

Interpreting Graphs

• Zero slope on a position-time graph: object is at rest.

• Positive slope: moving in the positive direction.

• Negative slope: moving in the negative direction.

• On a velocity-time graph, positive values indicate motion in the positive direction, negative values indicate motion in the negative direction.

Example: Determining Motion from Graphs

• Average velocity is the slope between two points on a position-time graph.

• Instantaneous velocity is the slope of the tangent at a specific point.

• Acceleration is the slope of the velocity-time graph.

Summary Table: Scalars vs. Vectors

Quantity	Type	Example	Direction?
Displacement	Vector	5 m east	Yes
Distance	Scalar	5 m	No
Velocity	Vector	10 m/s north	Yes
Speed	Scalar	10 m/s	No
Acceleration	Vector	2 m/s2 downward	Yes

Key Takeaways

• Always distinguish between scalar and vector quantities.

• Use sign conventions to indicate direction in one-dimensional motion.

• Apply kinematic equations for constant acceleration and free fall.

• Interpret motion using graphical methods for deeper understanding.

Additional info: These notes expand on the provided slides by including standard definitions, formulas, and examples commonly found in introductory college physics textbooks.