PHYS2211Lesson3

Chapter 3: Motion Along a Straight Line

3.1 Position, Displacement, and Average Velocity

Learning Objectives:

• Define position, displacement, and distance traveled.

• Calculate total displacement given position as a function of time.

• Determine total distance traveled and calculate average velocity.

Position

Position (x): Position refers to the location of an object at a specific moment, measured relative to a defined reference frame—commonly Earth. The position can be expressed in various units such as meters or kilometers.

• Importance of Reference Frames: The choice of a reference frame can significantly affect how position is perceived; for example, a person standing in a moving train will measure positions differently from someone standing on the ground.

• Example: A rocket’s launch can be described using a position function with respect to Earth's surface. Similarly, a cyclist’s position can be described relative to nearby landmarks, allowing for movement analysis.

Displacement

Displacement (Δx): Displacement captures the change in position of an object, represented as a vector quantity, which involves both magnitude and direction. It is calculated as: Δx = x_f - x_0, where:

• x_f is the final position,

• x_0 is the initial position.

• Displacement can be positive (indicating movement in one direction) or negative (indicating movement in the opposite direction).

• Example: If an object moves 2 m to the right and then 4 m to the left, the total displacement would be -2 m, indicating a net movement to the left.

Average Velocity

Average velocity is defined as the total displacement divided by the total time for that displacement: v_avg = Δx/Δt. This scalar value provides insight into an object’s overall motion.

• Example of Velocity Context: For Jill’s flyer deliveries, calculating average velocity involves considering her total displacement across various segments of her trip divided by the total time taken throughout the deliveries.

3.2 Instantaneous Velocity and Speed

Learning Objectives:

• Explain the differences between average and instantaneous velocity.

• Describe how velocity differs from speed.

Instantaneous Velocity

Instantaneous velocity is defined mathematically as the limit of average velocity as the time interval approaches zero: v(t) = lim(Δt → 0) (x(t + Δt) - x(t)) / Δt = dx(t)/dt.

• This concept is fundamental in physics because it provides a precise measure of how fast an object is moving at any given moment.

• Graphical Representation: It can be depicted as the slope of the tangent line drawn to a position vs. time graph at a specific time.

Speed

Speed is a scalar measurement of how quickly an object moves, irrespective of direction. It is calculated as: s_avg = Total distance / Elapsed time. When plotting speed, the graph does not consider the direction of movement, contrasting with velocity.

• Instantaneous Speed vs. Instantaneous Velocity: While instantaneous speed gives magnitude only, instantaneous velocity includes directional information, making it vital for thorough motion analysis.

3.3 Average and Instantaneous Acceleration

Learning Objectives:

• Calculate average acceleration and instantaneous acceleration.

Average Acceleration

Average acceleration represents the rate at which velocity changes over time and is expressed as: a_avg = Δv/Δt, with units in m/s². This calculation assists in understanding changes in an object's motion.

Instantaneous Acceleration

Instantaneous acceleration provides the acceleration of an object at a specific point in time, expressed as: a(t) = dv(t)/dt. This measure relates to the slope of the velocity vs. time graph, indicating how swiftly an object's velocity is changing at any moment.

3.4 Motion with Constant Acceleration

Key Equations for Constant Acceleration:

• Displacement: [ x = x_0 + v_0 t + (1/2) at² ]

• Final Velocity: [ v = v_0 + at ]

• Displacement from velocity: [ v² = v_0² + 2a(x - x_0) ]

3.5 Free Fall

Key Features:

Free fall refers to motion where gravity is the only acting force. All objects, irrespective of mass or shape, experience the same gravitational acceleration, approximately 9.81 m/s² when near Earth’s surface.

• Importance of Gravity: The equations of motion in free fall are specifically adapted to account for gravitational force, allowing for precise calculations in various contexts such as physics experiments and real-world applications like skydiving or projectile motion.

Example Problems

• Example 3.1: Analyzing Jill's total displacement and average velocity during various legs of her delivery trip utilizes incremental data derived from her position over time.

• Example 3.2: Deriving velocity vs. time from a provided position vs. time graph through calculation of slopes of segments.

Summary of Important Relationships

• Displacement (Δx): Change in position (x_f - x_0).

• Average Velocity (v_avg): Δx/Δt.

• Instantaneous Speed: Magnitude of instantaneous velocity.

• Average Acceleration (a_avg): Δv/Δt.

• Instantaneous Acceleration: a(t) = dv/dt.

• Free Fall: a = -g (gravitational acceleration).

This comprehensive overview encapsulates the essential concepts needed for understanding motion along a straight line and highlights relevant equations and examples, fostering a deeper comprehension of the subject.