PHYS 101 - LEC 2

Page 1: Introduction to Kinematics

• Always set up a coordinate system first:

  • Essential for correctly analyzing motion.

• Key concepts in kinematics:

  • Position: The location of an object in space.

  • Displacement: The change in position of an object.

  • Velocity: The rate of change of displacement.

  • Acceleration: The rate of change of velocity.

• Motion at constant acceleration is a key topic.

• Expectation: Familiarity with all material in Chapter 2 of the textbook.

Page 2: Example - Setting Up Coordinate Systems

• Scenario: A ball is thrown upward at 15.0 m/s from the edge of a cliff.

• Problem: Calculate the time for the ball to reach the base located 50.0m below.

• Coordinate system considerations:

  • Different options (A, B, C, D) for setting the reference frame.

Page 3: Example - Calculating Displacement and Velocity

• John's Movement:

  • Walked 70 m east and then 30 m west.

  • Total time: 50 seconds.

• Calculations required:

  • Displacement: Total distance in a single direction.

  • Distance: Total path length traveled regardless of direction.

  • Average speed: Total distance divided by total time.

  • Average velocity: Displacement divided by time.

Page 4: Average Speed vs. Average Velocity

• Question: Does average velocity convey the true speed of John?

  • Clarifies that average speed may obscure instantaneous variations.

• Improving average velocity understanding involves examining time intervals.

Page 5: Instantaneous Velocity

• Definition: The average velocity as the time interval approaches zero.

  • Mathematical representation: ( v = \frac{dx}{dt} )

• Interpretation: Slope of the tangent line on a position vs. time graph (x-t curve).

Page 6: Position vs. Time Graphs

• Instantaneous Velocity:

  • Determined by the slope of the tangent line at any point on the x-t curve.

• Average Velocity:

  • Calculated as the slope of a secant line connecting two points on the graph.

• Demonstration: Analysis using x-t and v-t graphs of a moving object on an inclined plane.

Page 7: Understanding Acceleration

• Definition: Acceleration is the rate of change of velocity.

• Average acceleration formula:

  • ( a_{avg} = \frac{\Delta v}{\Delta t} )

• Instantaneous acceleration:

  • Derivative of velocity with respect to time: ( a = \frac{dv}{dt} )

  • Indicates the slope of the v-t graph.

Page 8: Motion at Constant Acceleration

• Important formulas based on constant acceleration conditions:

  • Relate displacement, initial/final velocities, acceleration, and time.

• Question posed about the validity of formulas when acceleration varies over time.

Page 9: Falling Objects

• Key point: All objects near Earth's surface accelerate due to gravity.

• Acceleration due to gravity: ( g = 9.80 m/s^2 )

• Note: Coordinate system necessity if defining positive direction downward.

Page 10: Example - Thrown Ball

• Scenario analysis: Ball thrown upward at 15 m/s from a cliff edge.

• Calculation focused on time to reach 50 m below:

  • Various options for initial conditions presented (A-E).

• Coordinate system implications for problem solving.

Page 11: Example - Vehicle Collision

• Scenario: John driving at 126 km/h while a truck approaches at 90 km/h.

• Distance apart when brakes applied: 144 m.

• Questions:

  • Graphs required: Draw v-t and x-t graphs to analyze collision risk.

• If collision occurs, find relative velocity; if not, distance after stopping.

Page 12: Importance of Graphs vs. Equations

• Note: Graphs provide approximate solutions.

• Emphasis on the necessity of solving equations for accurate results.