Kinematics in One Dimension - Study Notes

PHYS131 Discipline of Physics - Kinematics in One Dimension

Chapter Overview

• Subject: Kinematics in One Dimension
• Author: Dr. Mathew K. Moodley
• Copyright: © 2014 Pearson Education, Inc.

Contents of Chapter 2

• Reference Frames and Displacement
• Average Velocity
• Instantaneous Velocity
• Acceleration
• Motion at Constant Acceleration
• Solving Problems
• Freely Falling Objects
• Graphical Analysis of Linear Motion

2-1 Reference Frames and Displacement

• Reference Frame: Any measurement of position, distance, or speed must be made with respect to a reference frame.

  • Example: If you sit on a train and someone walks down the aisle, their speed is a few miles/hour relative to the train, but much higher relative to the ground.

• Displacement vs. Distance:

  • Displacement: Denoted as $\Delta x = x<em>2 - x</em>1$, represents how far the object is from its starting point, irrespective of the path taken.
  • Distance: Measured along the actual path taken by the object.
  • Visual Representation: Displacement (solid blue line) vs. Distance (dashed line).

2-2 Average Velocity

• Speed: Defined as how far an object travels in a given time interval, measured in distance/time.
• Velocity: Incorporates directional information and is given by the formula:
  $v = \frac{\Delta x}{\Delta t}$
  Where $\Delta x$ is the displacement, and $\Delta t$ is the time interval.

Example 2-1: Runner's Average Velocity

• Position of a runner plotted over time along the x-axis:
  • Change in position from $x<em>1 = 50.0 m$ to $x</em>2 = 30.5 m$ over a $3.00 s$ interval.
  • Average Velocity Calculation = $\frac{x<em>2 - x</em>1}{\Delta t}$

Example Situation: Driving Home

• Driving steadily at $95 km/h$ for $180 km$, then slows to $65 km/h$ due to rain.
• Arrives home after $4.5 h$.
  • Questions: (a) Find distance to hometown; (b) Find average speed.

2-3 Instantaneous Velocity

• Instantaneous Velocity: The average velocity obtained in the limit as the time interval approaches zero.
• Illustrated by graphs showing constant velocity vs. varying velocity.

2-4 Acceleration

• Definition: Acceleration is the rate of change of velocity.
• Average Acceleration Formula:
  $a_{avg} = \frac{\Delta v}{\Delta t}$
• Given example at specific times:
  • At $t<em>1 = 0 s$, $v</em>1 = 0$,
  • At $t_2 = 1.0 s$, $v = 15 km/h$,
  • At $t_2 = 5.0 s$, $v = 75 km/h$.

Example 2-4: Average Acceleration of a Car

• A car accelerates from rest to $75 km/h$ in $5.0 s$.

  • Average acceleration is calculated using the defined formula from rest to final speed.

• Vector Nature of Acceleration:

  • In one-dimensional motion, only the sign (positive or negative) of acceleration is necessary.

• Distinction between:

  • Negative Acceleration: Acceleration in the negative direction as defined by the coordinate system.
  • Deceleration: Occurs when acceleration is opposite to the direction of velocity.

• Instantaneous Acceleration: Similar to instantaneous velocity, defined as the limit of average acceleration as the time interval approaches zero.

2-5 Motion at Constant Acceleration

• Average Velocity during a time interval $t$:
  $v<em>{avg} = \frac{v</em>0 + v}{2}$
• Constant Acceleration Equations: $v = v<em>0 + at$$x = v</em>0t + \frac{1}{2}at^2$$v^2 = v<em>0^2 + 2a(x - x</em>0)$
  • Equations relate to various variables and can be arranged as needed.

Example 2-7: Runway Design

• Designing for an airplane that requires a takeoff speed of $27.8 m/s$ with an acceleration of $2.00 m/s^2$.
  • Questions: (a) Can it reach required speed on a 150 m runway? (b) Determine minimum runway length if not.

Example 2-8: Car Acceleration

• A car accelerates from rest at a constant $2.00 m/s^2$ to cross a $30.0 m$ wide intersection. Calculate the time taken.

2-6 Solving Problems

1. Read and comprehend the problem statement.
2. Identify objects under study and time intervals involved.
3. Create a diagram and select coordinate axes.
4. List known quantities and the unknown ones that need to be found.
5. Determine applicable physics principles and plan a solution.
6. Identify relationships between known and unknown quantities. Verify equation applicability.
7. Solve algebraically for unknowns, checking results for dimensional correctness.
8. Calculate and round the solution appropriately.
9. Evaluate final result for reasonableness and confirm units.

2-7 Freely Falling Objects

• Near Earth's surface, all objects experience the same acceleration due to gravity (approximately $9.80 m/s^2$).
• In the absence of air resistance, all objects fall with this uniform acceleration.

Practical Example: Dropping Objects

• Scenario: A ball dropped from a $70 m$ high tower. Questions: (a) Time taken to reach the ground? (b) Velocity upon impact?

Further Scenarios

• A boy throws a ball upwards with an initial speed of $15 m/s$. Questions include:
  (a) Maximum height reached?
  (b) Time for ball to return to his hand?
  (c) Velocity upon return?
  (d) Time to reach $8 m$ above his hand?

2-8 Graphical Analysis of Linear Motion

• The slope of an x vs. t graph indicates velocity.
• Graphs illustrate varying velocity vs. time and resulting displacement curves.
  • Instantaneous velocity at any point is the tangent to the x vs. t curve.

Displacement Area Under v vs. t Curve

• The displacement of an object can also be calculated by determining the area under the velocity vs. time curve.

Summary of Chapter 2

• Kinematics: The study of motion in relation to a reference frame.
• Displacement: Defined as the change in position.
• Average Speed: Distance traveled/time elapsed; contrasted with average velocity which is displacement/time.
• Instantaneous Velocity: Defined as the limit of velocity for an infinitesimally small time period.