Crossfire off South Shore - BOCA race 2015. Yacht navigation requires a good understanding of velocity, distance and time.
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Home >> AP Physics I >> Kinematics
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Dynamics >>
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Kinematics is the study of motion. Specifically, just the mathematical description of the motion without considering what causes the motion. The latter study is known as dynamics. The father of kinematics was Galileo Galilei, and this formed the foundations that Isaac Newton built the science of dynamics upon. At AP level we concern ourselves purely with motion that is undergoing uniform acceleration. To consider anything further requires the use of calculus, which is not covered in this course.
We begin by analysing one dimensional motion and formulating the three main equations of motion. Once the students are comfortable solving problems, we move on to two dimensional motion, specifically projectiles.
We begin by analysing one dimensional motion and formulating the three main equations of motion. Once the students are comfortable solving problems, we move on to two dimensional motion, specifically projectiles.
The biggest jump from IGCSE to higher level physics (IB, A level or AP) starts here, when students realise that problem solving is no longer a simple matter of choosing an equation, rearranging it and sticking the numbers in to get the answer. The problems need to be planned out carefully and frequently require intermediate steps to tackle them. The key to this is usually a carefully drawn diagram and an understanding of what is actually happening, i.e. is the object accelerating of decelerating. This is the beginning of a process that we call 'modelling'. A valuable skill to have for many careers!
Unit 1 Student Workbook Updated: Jan 2017
Unit 1 - Kinematics Revision Questions
1.1 - Variables and Units
Objectives:
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Any quantity in science that can have a different value is called a VARIABLE. Usually in physics a variable can have a numerical value e.g. the mass of the boat is 200. It is critically important to state what the 200 means. There is a huge difference between a 200 g boat and a 200 ton mega yacht. A navigator trying to determine the boat's arrival time at a port must know whether the boat's speed is 12 km/h, 12 mi/hr, 12 m/s or 12 knots. Strictly speaking, an answer in physics that does not explicitly state the unit of measurement is wrong. For this class, I will be deducting 1/2 a mark for missing off the unit. If the unit is wrong, then the answer must of course be wrong!
There are many different units in the world. They depend on the context, who defined them, the scale that you are dealing with and sometimes the nationality of the problem. The International Standard system of measurement, Systeme Internationale (SI) or metric, was initiated by Napoleon in the 18th century due to a frustration between the measurement systems of the various countries and even regions that he conquered. The scientific community and the AP exam use exclusively the metric system. The US and Bermuda still use the antiquated Imperial or 'Standard' or 'British System' of measurements in everyday life. So, occasionally you will have to convert them. Conversion tables are readily found online or inside the back cover of your text book. You are not be expected to learn the conversions between Imperial and Metric.
The metric system uses a set of prefixes for convenience. You are expected to be fluent in these before the exam as many equations only work if you use standard units, e.g. you may need to convert a length of 24 cm to 0.24 m. Converting the units to their base level at the start of a problem is a good habit to get into right from the start of the course.
There are many different units in the world. They depend on the context, who defined them, the scale that you are dealing with and sometimes the nationality of the problem. The International Standard system of measurement, Systeme Internationale (SI) or metric, was initiated by Napoleon in the 18th century due to a frustration between the measurement systems of the various countries and even regions that he conquered. The scientific community and the AP exam use exclusively the metric system. The US and Bermuda still use the antiquated Imperial or 'Standard' or 'British System' of measurements in everyday life. So, occasionally you will have to convert them. Conversion tables are readily found online or inside the back cover of your text book. You are not be expected to learn the conversions between Imperial and Metric.
The metric system uses a set of prefixes for convenience. You are expected to be fluent in these before the exam as many equations only work if you use standard units, e.g. you may need to convert a length of 24 cm to 0.24 m. Converting the units to their base level at the start of a problem is a good habit to get into right from the start of the course.
A note regarding formatting. When typing variables and units it is easy to get them muddled as the same letters are often used in different contexts. The standard is that variables are written in italics and the unit in normal case with a space after the value. E.g. the symbol for mass is either M or m, while the symbol for a metre is also m and mega is M! In practice this is not as confusing as it appears. When you write these by hand, the context is determined solely by the location of the symbols.
Try to follow the teacher's formatting and don't randomly change between lower and upper cases.
Try to follow the teacher's formatting and don't randomly change between lower and upper cases.
m = 13.5 kg and l = 3.2 m.
Trivia Question: Which very commonly used variable has never successfully been given a metric unit?
Physics uses a lot of equations! These follow the basic rules of algebra that you have studied in maths class. It is expected that you are totally fluent in algebra. After all: algebra was sort of invented to solve physics problems...
Top tip - if there is a square root in the equation, get rid of it by squaring everything before trying to rearrange it.
Top tip - if there is a square root in the equation, get rid of it by squaring everything before trying to rearrange it.
ACTIVITY - Discuss the above cartoon!
1.2 - Problem Solving
Objectives:
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Without a shadow of a doubt, the hardest aspect for any student is to try and solve a physics problem! Any student can learn an equation, rearrange it and plug in the known variables and churn out an answer. You will never see questions like this on an AP or university exam! The hard part is to figure out how to tackle the problem and many students starting out get stuck. The method that I use, and STRONGLY ENCOURAGE is to draw a well-labeled diagram of the problem, with everything that you know marked on it. This is the beginning of the process that we call MODELLING, and that is a highly transferable skill that is used in science, engineering, maths, weather and climate, finance and insurance.
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An example of a well-labelled diagram for a kinematics problem. Note that all the known variables are clearly shown, as it what the unknown variable that we are trying to find (the velocity after 5 seconds. You could also label the time as t = 5 s between the pictures of the cars. Top Tip: make your diagrams large enough that you can add to them afterwards if you need to.
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| cw_0.1_problem_solving_challenges.pdf |
1.3 - The Kinematic Variables
Objectives:
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With kinematics problems, if you know any THREE of the five variables, you can calculate the other two. So, the trick is to identify them when you draw the diagram. Some students prefer just to list the variables that they know. Although I am not a fan of this practice, it works for simple scenarios. Sometimes the variables are not explicitly mentioned in the problem. Look for the clues: At rest or stationary means that the velocity at that point is zero. When an object is thrown into the air, the vertical velocity at the maximum altitude is zero. The acceleration of free fall is g. You should also be confident that you can measure each variable in the lab using the most accurate methods available and describe the experiments in detail. Problems at this level often consist of multiple parts - so the variables can 'change' names - e.g. the final velocity of the first part of the motion will become the initial velocity of the second part. We will also cover v-t graphs, so you should be able to identify the five variables on any given graph. At AP level the acceleration for any given segment of an object's motion will be constant.
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from Graphing Motion - interactive figures that are related to the College Physics textbook that we use. This is one of the freely available ones from Wadsworth Media.
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Graphing Motion in 1-dimension. Simulation by Walter Fendt
This is rather basic but is really good to show the differences between x-t, v-t and a-t graphs. The concept of the gradient representing the rate of change should be clearer after playing with this a few times. |
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Physics Aviary Game: Match the graphs to the motion.
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Couple of YouTube videos from the coolest youtube teachers in the US. Professor Dave and Mr Anderson. They have a wide range of high quality videos and if you haven't understood what I have been talking about in class they are well worth the time.
Khan Academy Tutorial - Acceleration
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VideoPhysics - short instructional video to the motion analysis app that we will be using a lot over the course.
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1.4 - The Equations of Motion
Objectives:
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There are three basic linking equations between the kinematic variables. Textbooks often have another one or two, but these three will do everything and it is well worth the effort to put them to memory either by learning them rote or just lots of practice. You should be familiar with their derivation although that is not required in the exam.
The first equation should be familiar to you. It is the acceleration and the equation is based on the gradient of the v-t graph.
The first equation should be familiar to you. It is the acceleration and the equation is based on the gradient of the v-t graph.
The second equation is based on the area under the v-t graph. Students often struggle with rearranging this at first as it almost has everything: we have fractions, addition/subtraction/multiplications and squares. Practice! Frequently the initial velocity is set to zero, which simplifies things a great deal.
Another fun one to rearrange if neither of the velocities are set to zero.
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You should become confident that you can use these equations in conjunction with lab work. With some of the one-dimensional problems that you have been set in class and for homework you may have to either a) break a scenario into two of more segments or b) be really careful with the signs of the vectors. A classic problem is: you are standing on the edge of a cliff and throw a ball upwards. How fast will it hit the ground? This is best thought of split into TWO parts. Part 1: The flight of the ball upwards from the edge of the cliff to it apex of flight. Part 2: The free fall descent of the ball. However, if you are careful, you can solve the whole problem in one go as the acceleration is constant throughout the scenario. Now: if the ball had a rocket engine attached to it first, then you MUST 'divide in order to conquer'.
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1.5 - Two Dimensional Motion
Objectives:
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This is where the fun really starts! So far, we have only considered one-dimensional motion. But in everyday life we have two and three-dimensional motion all the time. At AP level we only consider two-dimensions, but you could extend this to three-dimensions using the same basic ideas. The standard scenario for this sort of work at AP is projectile motion (ballistics).
KEY CONCEPT: Vectors that are perpendicular to each other can be treated independently. Time is the linking variable.
The image above shows a ball following a parabolic ballistic trajectory. Consider both the vertical and horizontal motions.
Vertical: The ball decelerates on the way upwards to a stop at the apex and then accelerates downwards again. The acceleration is g.
Horizontal: The ball is moving at a constant speed as evinced by the equal spacing of the lines.
The time of flight is the same - the ball only moves sideways for as long as it is in the air. The time is also symmetrical. The time up = time down.
(Note: air resistance is always neglected in these situations.....)
You will be performing a number of lab experiments on this topic.
Historical trivia: this is one of the things that made mechanics respectable in the early days! Once physicists could accurately predict the range of a cannonball, lords and kings started to take notice.
Vertical: The ball decelerates on the way upwards to a stop at the apex and then accelerates downwards again. The acceleration is g.
Horizontal: The ball is moving at a constant speed as evinced by the equal spacing of the lines.
The time of flight is the same - the ball only moves sideways for as long as it is in the air. The time is also symmetrical. The time up = time down.
(Note: air resistance is always neglected in these situations.....)
You will be performing a number of lab experiments on this topic.
Historical trivia: this is one of the things that made mechanics respectable in the early days! Once physicists could accurately predict the range of a cannonball, lords and kings started to take notice.
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Example 1
The simplest problem is the most common and involves firing a projectile horizontally from a height. The initial vertical velocity is zero and the initial horizontal velocity is the launch speed, v. Step 1 - calculate the time-of-flight from the vertical motion. as initial vertical speed is zero and acceleration is g, this becomes:
Step 2 - calculate the range from the horizontal motion.
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Example 2
If the ball is thrown upwards at angle (as shown above) - then we need split the velocity into vertical and horizontal components before we do any physics. This is where sines and cosines come in: Looking carefully at the diagram you should be able to see that the horizontal velocity vector is constant, while the vertical one changes continuously. The actual speed of the ball can be found by applying Pythagoras's Theorem.
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So, the first step in determining the range of the projectile is to find out the time-of-flight. This is done by finding how long it takes to go up and fall back down again. We usually know three variables: the initial vertical speed, the acceleration (gravity) and the final vertical speed. So, using the first of the kinematic equations gives us:
This is the time to go up. The total time-of-flight is double this, so we have:
Step 2: Consider the horizontal motion. The ball is traveling at a constant speed of the horizontal component of the initial speed. *can't do subscripts yet*
With a bit of tidying up becomes:
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Example 3
This is the most challenging of all the situations and requires all of the above and dividing the flight into two distinct sections: up and down. It is above the standard required for AP - however, you may be asked to either a) describe the effects of raising or lowering the launch height or b) changing the angle of the launch. Anyway, the range is below. Advanced students can see if they can derive it! These screenshots from projectiles lab work carried out 22 Oct by AP Physics 1 class using the VideoPhysics app on an iPad. As can be seen on the left, it is not always sunny in Bermuda! |
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PhET Simulation - Projectiles. A new and improved simulation from the Colorado team. The advantage of the new version is that it is much easier to adjust the height of the cannon.
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Challenge: Can you solve the problem set by Peter's teacher?
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Physics Aviary Game: Drone Delivery Challenge. The idea to drop packages from a moving drone onto a target - Amazon Prime??
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Crazy photo that went around the internet a few years ago. What is happening!?
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Other Resources
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PhET Simulation that shows how a vector can be broken down into two perpendicular components.
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Bozeman Science - Motion This is an excellent video that describes position, velocity and acceleration
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Motion Graphs - interactive graphing
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Wonders of Physics - Kinematics. Excellent cartoons from Evan Toh
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CK12 Physics - Motion - great site loaded with videos
Interactive projectiles simulation - fun one with sound effects!
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Tracker - a computer based version of the VideoPhysics app. A bit more flexible than the app and more accurate as you can import a video from a higher quality camera than the iPad has. The mouse pointer is easier to use! The video tutorial is a bit of a must.
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