Physics 30 Complete Course Study Booklet

Organized Notes Compiled by Anthony Demong from Mr. Christoffel’s Physics 20 Class and Resources

NOTE: This guide does not contain absolutely everything learned in the course, though it covers the most important material. It is designed to accompany questions from the textbooks provided by the teacher.

Kinematics Unit – Study Guide Compiled by Anthony Demong

Definitions and Important Notes

Kinematics: Kinematics is the branch of physics that deals with the description of the motion of objects without reference to the forces or agents that cause said motion. The branch that does deal with this is called dynamics.

Displacement: Displacement describes both the magnitude of a change in position and the direction of that change. If a dog walked 20.0 metres east of its original starting position, it would have a displacement of 20.0 m [E] (20.0 metres is the magnitude of the change in position, and the [E] represents the direction east).

Delta (Δ): Delta is a Greek letter that represents “change of”, “increase of”, “decrease of”, “difference”, or “interval” (e.g. Δt would represent ‘change in time’).

Vectors / Vector Quantities: A vector quantity is a quantity that has a magnitude, unit and direction. Displacement is a vector.

Vectors can be represented graphically as arrows of varying length, according to a certain scale. The following arrow would represent a dog’s displacement 20.0 m [E]:

Scale: 1 cm = 10.0 m

A vector should also have an arrow on top of its variable to indicate that it is a vector:

Remember:

  • means displacement = the final position – the initial position
  • If the value of the vector is negative, draw the arrow facing the other direction

Scalars: Scalars are quantities that only have a unit and magnitude. Time, mass, and length are scalars.

Uniform Velocity / Uniform Motion: Uniform velocity is a constant velocity that does not accelerate or decelerate over time. It can be found using

Average Velocity: Average velocity can be calculated using the formula

It can also be found by adding all the velocities, then dividing that sum by the number of different velocities.

Instantaneous Velocity: Instantaneous velocity is the same as average velocity if there is uniform velocity. If there is not uniform velocity, then the instantaneous velocity will differ from the average velocity.

In a displacement vs time graph, instantaneous velocity can be found by making a tangent line to the line of best fit, then creating a triangle from the two ends of the tangent line (so the tangent line is the hypotenuse of the triangle).

Based on the two endpoints of the tangent line, instantaneous velocity can be calculated using the formula

Acceleration: Acceleration is the rate of change of velocity. Constant acceleration on a time vs displacement graph appears as a sloping line, while constant acceleration on a time vs velocity graph appears as a straight line.

Acceleration can be found using the formula or

Average Acceleration: Average acceleration can be calculated with the formula

Instantaneous Acceleration: Instantaneous acceleration is the same as average acceleration if there is constant acceleration. If there is not constant acceleration, then the instantaneous acceleration will differ from the average acceleration.

In a velocity vs time graph, instantaneous acceleration can be found by making a tangent line to the line of best fit, then creating a triangle from the two ends of the tangent line (so the tangent line is the hypotenuse of the triangle).

Based on the two endpoints of the tangent line, instantaneous acceleration can be calculated using the formula

Other Important Formulas and Their Uses

  • can be used to calculate the final velocity based on the initial velocity, time, and acceleration
  • can be used to calculate the displacement based on the initial velocity, time, and acceleration
  • represents that the difference between the final velocity squared and the initial velocity squared is equivalent to twice the acceleration multiplied by the displacement

Position over Time Graphs

When graphing, remember the following (specific to Mr. Christoffel’s class):

  • Darken the x-axis and y-axis of the graph to emphasize them
  • Create a meaningful and relevant title for the graph
  • Label the axes with their proper units (e.g. cm, s, etc.)
  • Use the proper number of significant figures on your axes
  • When labelling plotted points on the graph, use the same number of significant figures for the x and y values
  • Draw a circle around any points to indicate uncertainty
  • Any line of best fit drawn onto the graph should have roughly the same number of points on either side of it
  • To find the slope, find two points on the line of best fit and use them to draw a triangle
  • Determine the slope of the triangle’s hypotenuse, remember to take account for the proper number of significant figures in the calculations

Example:

*Don’t make the same mistake that I did. Always state the formula before making calculations, and do not put instead

Kinematic Graphing of Displacement, Velocity and Acceleration

  • Always try to picture a mental image of an object in motion when working on these graphs to assist you in making sense of the increases and decreases in displacement, velocity and acceleration
  • Set up the graph just like the position over time graph exemplified above, but split the graph into three rows: one for displacement, one for velocity, and one for acceleration
  • First, outline the time increments in the problem: For the problem below, those would be 10.00 seconds then an added 3.50 seconds
  • Draw vertical lines in the graph along those times to split it into several regions; you may want to label those regions as A, B, C, etc.
  • After that, solve for as much data possible:
  • The initial velocity / displacement
  • The final velocity / displacement

(You may need to stop and think about whether or not the object is moving or resting at the beginning or the end)

  • The velocity / acceleration at each time increment
  • Graph the data, and shade in the acceleration and velocity’s areas to help with calculations. Remember to use the proper number of significant figures

Example:

In this graph, a tractor is moving at a velocity of 12.00 metres per second for 10.00 seconds, then decelerates to a velocity of 7.00 metres per second in a time of 3.50 seconds.

River Crossing Problems

A river has a current with a velocity of 4.0 metres per second north. A swimmer can swim at a velocity of 6.0 metres per second in the water east. What is the velocity of the swimmer compared to the ground, and in which direction?

Drawing the Diagram

Before the vector diagram is drawn, a basic diagram of the situation should be drawn to represent the scenario. Below, a swimmer is swimming east across a north-flowing river, and is being pushed north as he swims in a northeast direction:

By drawing this essential scenario diagram, we know the basic directions of each vector, and that he will be travelling northeast when compared to the ground.

Next, the vector diagram can be drawn. Each arrow is drawn in scale to each velocity in the diagram (i.e. the water current vector arrow is drawn to the scale of 4), and in the same direction (the river velocity is drawn pointing north (up), the swimmer velocity is drawn pointing east (left), and the resultant velocity is drawn pointing northeast (up and to the left) :

REMEMBER TO LABEL IT WITH THE VECTOR EQUATION hypotenuse = side + side

Finding the Resultant Vector

To calculate the resultant vector’s magnitude, use Pythagorean Theorem:

  • Remember to remove the vector signs () and replace them with exponents (2)
  • You may need to rearrange the equation if you are solving for a specific side, and not the hypotenuse

To calculate the resultant vector’s direction, use COS SIN TAN based on the vector diagram drawn above:

Therefore the swimmer is swimming at a velocity of 7.5 metres per second 34° north of east.

Projectile Problems

  • Whether a projectile is projected horizontally or vertically, it will take the same amount of time to fall (both things will hit the ground at the same time)
  • Since vertical velocity is accelerated, in the equation replace with (height), and with ([down]) to make
  • Since horizontal velocity is uniform, in the equation replace with (range) to make

Circular Motion Problems

Equations

Variables

  • = Velocity of an object in circular motion
  • = Amount of time necessary to make one complete oscillation around the circle
  • = Acceleration of an object in circular motion
  • = Radius of the circle

Notes

  • Centripetal Acceleration () is a force acting an object in circular motion that prevents it from flying out of the circle. In the diagram below, it would be an arrow pointing into the centre of the circle

Diagram

Adding Vectors that Do Not Form a Perfect Right Triangle

The steps will be narrated below in an attempt to explain the process, because heck, I’m not even sure if the title is the right way to describe it.

Problem

A person walked 8.94 km @ 27.0° north of west, then 11.40 km @ 76.0° north of east. Find his total displacement.

Variable Setup

First, the angles must be converted into standard angle format. This is simple converting the angles based on a system where east becomes 0°, north is 90°, west is 180° and south is 270°.

Diagram Stage

The first process is to draw each vector, connected in order. Then, the x and y values of each vector are drawn, a sum of both the x and y vectors are drawn, and the resultant vector is drawn connecting the tail of one vector to the tip of the other:

Calculation Stage

Then, a reference triangle is drawn (don’t worry about magnitudes or directions) to find whether COS, SIN, or TAN would find help to find the sum of the x and y values:

REMEMBER TO ALWAYS REMOVE THE VECTOR SIGN () WHEN DIVIDING VECTORS

Next, calculate the sum of both the x and y values of each vector like shown below:

Next, draw the vector diagram of the sums of the x and y values for each vector to find the resultant displacement.

N of E

Therefore his total displacement was 10.83 kilometers 83.4° north of east.

Dynamics Unit – Study Guide Compiled by Anthony Demong

The following unit relies heavily on the knowledge gained from the kinematics unit. Ensure that you have a comprehensive understanding of kinematics before attempting to solve dynamics problems.

Definitions and Important Notes

Dynamics: Dynamics is the branch of physics that deals with the description of the motion of objects with reference to the forces or agents that cause said motion.

Unbalanced Force:Any force that does not have an equivalent and opposite force acting upon it at the same time. Balanced forces are known as statics.

Newton’s 3 Laws:

  • A body in motion remains in motion unless an external unbalanced force acts upon it
  • The relationship between an object’s force (F), mass (m) and acceleration (a) is represented by the formula
  • For every action there is an equal and opposite reaction

Acceleration: Acceleration is the rate of change of velocity.

Force: Force is measured in newtons (N). One newton is the force required to accelerate 1 kilogram 1 metre per second. Weight is a measurement of force, since it depends on both mass and gravitational acceleration.

Mass: A measurement of matter. In this class it is most often measured in kilograms (kg).

Net Force: Represented by, the net force of an object is determined by finding the sum of the vectors of the applied forces and force of friction:

Force of Gravity: The force of gravity is the force that an object has in the direction of the Earth’s surface. On Earth, the acceleration of gravity is 9.81 metres per second, so the force of gravity acting upon an object can be determined by multiplying the acceleration of gravity by the mass of said object (using Newton’s second law of ()).

Angle of Inclination: The angle of inclination is the angle in which a slope is measured. For example, a vertical wall would have an angle of inclination of 90°, while a flat plane would have an angle of inclination of 0°.

Normal Force: The normal force is the force acting upon an object perpendicular to the surface that it is in contact with:

The normal force can be calculated using the formula

Downwards Force:Downwards force, represented by is the force of an object in the direction of its travel down a slope. It is opposite to the force of friction and parallel to a surface:

The downwards force can be calculated using the formula. However, it can also be calculated using the formula.

Sample Calculation

A force of 120N is applied to a mass of 3.50kg. What is the acceleration?

F = 120N

m = 3.50kg

a =?

∴ The mass had an acceleration of 3.43 metres per second squared.

μ – The Coefficient of Friction

The letter μ can be used to represent the coefficient of friction. The coefficient of friction is a ratio of the force of friction compared to the normal force.

QUICK TIP: Never use vector signs () when dividing vectors. Always remove them from the dividend and divisor when putting them into a formula.

The diagram below represents a mass of 13.0kg on a slope with an angle of inclination of 30.0°.

The normal forceis acting upon the mass perpendicular to the surface. The object’s force of friction is acting against the downwards force, both of which are parallel to the surface. The force of gravity is acting upon the object downwards towards the Earth. It is also very important to notice that the right triangle formed between the normal force, force of gravity and downwards forcehas the same angle as the angle of inclination.

SOH CAH TOA can be used to determine the value of the angle of inclination, downwards force, and force of gravity if needed.However, remember to show all of the steps when determining anything from SOH CAH TOA.

Pushing and Pulling Objects

Whenever an object is pushed or pulled, multiple forces are acting upon said object. When an object is pushed, downwards force is also often exerted onto the object which can increase friction. When an object is pulled, upwards force is also often exerted into the object which can decrease friction.

To begin the solving process for this problem, list all of the knowns and unknowns:

Step 1: Solve for Fappx

Step 2: Solve for Fappy

Step 3: Solve for Fnety

 Converts to negative because gravity is working against the pulling force. If the object was pushed gravity would be added.

Step 4: Solve for Fnetx

 The force of friction is equivalent to μ multiplied by the normal force, which in this case is Fnety.

Step 5: Solve for anetx

Remember: The difference between push and pull questions is whether or not the force of gravity is added or subtracted from the applied force in the y direction.

Tension and Pulleys

The diagram below represents three masses on a surface, two of which are hanging from pulleys. The larger mass (m1) is affected greater by the force of gravity and is therefore pulling the weaker hanging mass (m3), as well as the mass on the surface (m2) towards it.

For ease, the scenario can also be represented like above, where the masses are all on a single flat surface. To calculate the net acceleration in the x direction (), the net force in the x direction () must be found (if m1=250g, m2=500g andm3= 125g):

The tension for each mass can then be found be rearranging the equation for Fnet(demonstrated with m1):

The net acceleration in the x direction is the same for every mass, so it can be used here

The Universal Gravitational Constant

The universal gravitational constant (G) is equivalent to . It can be used in the following formula to determine the effects of gravity on other celestial bodies:

The above formula states that the force of gravity (on another celestial body) is equivalent to the universal gravitational constant multiplied by mass 1 (the central mass, in this case the mass of a celestial body) multiplied by mass 2 (the mass of the object being pulled towards the celestial body) divided by the distance between the object and celestial body squared.

Note: The subject of the formula (mass 1) does not need to be a celestial body.

Momentum and Energy Unit – Study Guide Compiled by Anthony Demong

The following unit relies heavily on the knowledge gained from both the kinematics unit and the dynamics unit. Ensure that you have a comprehensive understanding of kinematics and dynamics before attempting to solve any momentum or energy problems.

Definitions and Important Notes

Momentum: Momentum is the quantity of motion of a moving body. It is the product of the object’s velocity and mass:

When working with momentum questions, use vectors. It is measured in kg/ms, or Ns (since kg/ms2 is one newton).

Impulse: Impulse is the change in momentum. Like momentum, it is measured in kg/ms or Ns. It is also the product of force and time:

Conservation of Momentum: Momentum is conserved when the mass or velocity of an entity is changed. There are three main types of collisions concerned with the conservation of momentum in our class:

  • Inelastic, where two objects collide and stick together
  • Elastic collision, where two objects collide and bounce of each other
  • Perfectly elastic collision, where two objects collude and bounce off each other in such a way that all energy is conserved

The sum of the initial momentums of both colliding objects is equivalent to the sum of their final momentums: