PHYS 101 - LEC 3

Page 1: Introduction to 1-D Kinematics

• Focus on 1-dimensional kinematics.

Page 2: Average Velocity - Swimmer A

• Initial Position:

  • At time ( t_1 = 0 ), position ( x_1 = -10.0 \text{ m} )

  • At time ( t_2 = 25.0 \text{ s} ), position ( x_2 = 40.0 \text{ m} )

• Calculations:

  • Displacement:

    • ( \Delta x = x_2 - x_1 = 40.0 \text{ m} - (-10.0 \text{ m}) = 50.0 \text{ m} )

  • Time Elapsed:

    • ( \Delta t = t_2 - t_1 = 25.0 \text{s} - 0 = 25.0 \text{s} )

  • Distance (length of pool):

    • ( d = 50.0 \text{ m} )

  • Average Velocity:

    • ( v_{avg} = \frac{\Delta x}{\Delta t} = \frac{50.0 \text{ m}}{25.0 \text{ s}} = 2.00 \text{ m/s} )

  • Average Speed:

    • ( v_{avg (speed)} = \frac{d}{\Delta t} = \frac{50.0 \text{ m}}{25.0 \text{ s}} = 2.00 \text{ m/s} )

Page 3: Average Velocity - Swimmer B

• Initial Position:

  • At time ( t_1 = 0 ), position ( x_1 = 40.0 \text{ m} )

  • At time ( t_2 = 100.0 \text{ s} ), position ( x_2 = -10.0 \text{ m} )

• Calculations:

  • Displacement:

    • ( \Delta x = x_2 - x_1 = -10.0 \text{ m} - 40.0 \text{ m} = -50.0 \text{ m} )

  • Time Elapsed:

    • ( \Delta t = t_2 - t_1 = 100.0 \text{s} - 0 = 100.0 \text{s} )

  • Distance:

    • ( d = 50.0 \text{ m} )

  • Average Velocity:

    • ( v_{avg} = \frac{\Delta x}{\Delta t} = \frac{-50.0 \text{ m}}{100.0 \text{ s}} = -0.500 \text{ m/s} )

  • Average Speed:

    • ( v_{avg (speed)} = \frac{d}{\Delta t} = \frac{50.0 \text{ m}}{100.0 \text{ s}} = 0.500 \text{ m/s} ) (always positive)

Page 4: Meeting Point of Swimmers

• Swimmer A:

  • Position Function: ( x_A = -10.0 + 2.0t )

• Swimmer B:

  • Position Function: ( x_B = 40.0 - 0.50t )

• Graphical Method:

  • Plotting ( x_A ) and ( x_B ) vs. time to find the intersection (meeting point).

  • Approximate meeting time at ( t = 20 \text{s} ) and position ( x = 30 \text{m} ) found through graphical analysis.

Page 5: Exact Solution for Meeting Time

• Setting Positions Equal:

  • Solve for ( t ) when ( x_A = x_B ):

    • ( -10.0 + 2.0t = 40.0 - 0.50t )

  • Rearranging gives:

    • ( 2.0t + 0.50t = 40.0 + 10.0 )

• Final Calculation:

  • ( 2.5t = 50.0 \Rightarrow t = 20.0 \text{ s} )

  • Validate positions:

    • ( x_A = -10.0 + 2.0 \times 20.0 = 30.0 \text{ m} )

    • ( x_B = 40.0 - 0.50 \times 20.0 = 30.0 \text{ m} )

Page 6: Ball Thrown from a Cliff

• Scenario:

  • Ball thrown upward with speed ( 15 \text{ m/s} )

  • Cliff height: ( 50 \text{ m} ) below.

• Projectile Motion Specifications:

  • Initial Position ( x_0 = 50 , m )

  • Initial velocity ( v_0 = 15 , m/s )

  • Acceleration due to gravity ( a = -9.8 , m/s^2 )

• Clicker Question:

  • Setup to solve for time ( t ) until reaching the base: using kinematic equations.

Page 7: Collision Scenario - Vehicles

• Scenario Description:

  • John driving at ( 126 \text{ km/h} ) encounters a truck going ( 90 \text{ km/h} )

  • Both vehicles decelerate at ( 6.00 \text{ m/s}^2 )

  • Initial distance apart: ( 144 , m )

• Tasks:

  • A) Create velocity and position-time graphs to analyze collision likelihood.

  • B) If collision occurs, determine relative velocity at collision; if not, find stopping distance of both vehicles.

Page 8: Coordinate System Setup

• Setting Up:

  • Establish coordinate systems for both vehicles to simplify analysis.

Page 9: Kinematic Equation Overview

• Formula:

  • ( v = v_0 + at )

• Position Formula:

  • ( x = x_0 + v_0t + \frac{1}{2}at^2 )

• Graphs Visualization:

  • Sketching trajectories and velocities over time.

Page 10: Continued Analysis of Vehicle Collision

• Vehicle Motion Contributions:

  • Compare changes in position over time for accuracy in collision estimation.

  • Visualize the motion with graphs to ascertain the impact of deceleration.

Page 11: Position-Time Curves

• Graphs:

  • Important to illustrate the position changes of both vehicles over time to identify collision points visually.

Page 12: Vehicle Collision Outcome Analysis

• Detailed Time Analysis:

  • Report the position of each vehicle at specific time intervals leading up to their stop.

  • Example calculations at times ( t = 15, 25, 35 ) seconds.

• Collision Confirmation:

  • Based on time-dependent position analysis, confirm whether collision occurred.

Page 13: (No specific notes)

Page 14: Importance of Accurate Solutions

• Graphical vs. Analytical Solutions:

  • Emphasize the need for solving equations analytically for precision rather than relying solely on graphical estimates.

• Conclusion:

  • Always verify findings through systematic computation alongside visual aids.