These are my complete notes for Range and Relative Motion in Classical Mechanics.
I color-coded my notes according to their meaning - All numbered notes (which I call rules) are red, and include examples and the basis for understanding a topic. Definitions are written in green, and other important information (such as large-scale drawings that are better visualized than explained) was written in blue. All of this information is preserved on this page, with logical flow and breaks. I use ascii line drawings sparingly - If I can convey information or a graph using an image online, I will do so.
All of the knowledge present in these notes are filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes. In the unlikely event there are any egregious errors, contact me at jdlacabe@gmail.com.
Summary of Range & Relative Motion (Classical Mechanics)
IV. Range & Relative Motion.
IV.I Basics of Range.
P. Rule 35. Here is the 'Range' equation, giving you the horizontal displacement (which, you should note, is not written with a Δx, as that would displacement as a whole). There are specific characteristics of a question that will allow you to use the range equation, in the same manner as the U.A.M. equations. If a question explicitly says that an object in projectile motion is landing at the same height which it was thrown (or something to that effect), then the range equation can be used as the displacement in the y direction is zero.
Range = (Vi² × sin(2θi)) / g
R = Range = Δx
Vi = ||Vi|| (Magnitude of Vi)
θi = Initial Angle or Launch Angle
g = Acceleration due to gravity on Earth, POSITIVE 9.81 m/s²
Because both the magnitude and direction of the initial velocity vectors are present in the equation, the initial velocity does not need to be resolved into its components for use in the equation.
g: In projectile motion, ay = -g
g = +9.81 m/s²
ay = -9.81 m/s²
Given this information, we know that the value in the bottom of the equation is not the negative acceleration we are accustomed to, but rather the positive acceleration due to gravity on planet Earth, 9.81 m/s². Because gravity has the dimensions of m/s², the rest of the variables must use the standard meters and seconds for their dimensions, m/s for ||Vi|| and meters for the range itself.
θi: The sine of any angle can never be greater than one. Therefore, the maximum range an angle can have (given the magnitude of the velocity initial is constant, which is should considering this is the x-direction) is discoverable through simply setting the sine value of the equation equal to one:
1 = sin(2θi)
2θi = sin⁻¹(1) = 90°
θi = (90° / 2) = 45°.
Therefore, a launch angle of 45° will give us the maximum range of an object given a constant magnitude of the initial velocity. Any angle other than 45° will give a smaller range.
Furthermore, being familiar with the graph of sin(2θ), we know that any two complementary angles will have the same sine value and range, such as 30° and 60°. Mathematically, this is shown as sin(2θi) = sin(2(90 - θi)).
JUST BECAUSE TWO ANGLES HAVE THE SAME RANGE, THAT DOES NOT MEAN THAT THEY HAVE THE SAME PROJECTILE PATH. An object launched at 60 degrees will go higher than one launched at 30 degrees, but they will end up in the same place.
Memorization tool for Range:
I believe the best way to memorize new information is to associate it with something that we have already memorized that we subconsciously reinforce through repetition: music. Everyone has already memorized hundreds of melodies, so finding one that would match the rhythm and meter of the equation (and there is meter) is not too terribly difficult.
1. To the theme of Nessun Dorma:
Velocity initial squared, times sine 2 theta, initial, divided by gravity...
IV.II Frames of Reference.
P. Rule 36. Relative motion is done with vectors. Imagine a van and a motorcycle driving side by side: the velocity of the van relative to Earth is 24 mi/hr in the East direction. The velocity of the motorcycle relative to the Earth is 13 mi/hr in the East direction. These values would be written thusly:
VvE = 24 mi/hr E
VmE = 13 mi/hr E
In order to find the velocity of the van with respect to the motorcycle, we must use Vvm as the term. This value will be the difference between the two vectors.
VvE = VmE + Vvm
Vvm = VvE - VmE
Vvm = 24 mi/hr E - 13 mi/hr E
Vvm = 11 mi/hr E
You may notice that by taking the negative of the vector, we are in effect reversing its direction. This can be represented by switching the order of the subscripts.
-VmE = VEm
= -13 mi/hr E
= 13 mi/hr W
Therefore, from the point of view of the motorcycle, it would appear that the Earth is moving westward at 13 mi/hr. Finally, in order to solve the velocity of the motorcycle with respect to the van, we simply need to take the negative of the already known value of the motorcycle with respect to van:
Vmv = -Vvm
=-11 mi/hr E
11 mi/hr W
Thus, the van, which is moving faster than the motorcycle, would see the motorcycle as moving backwards (west) as 11 mi/hr.
P. Rule 37. There is a trick that can be used in all relative motion subscripts: any time you sum two relative motion velocities and they share a common subscript on the inside or the outside, that subscript will cancel out in a specific way. Using the example from Rule 36, we can understand what this specifically means:
Vvm = VvE + VEm
The "E" (Earth) on the inside of the two velocities will cancel out, and because the common subscript was on the inside, the subscripts will combine forwardly into vm.
VvE = VmE + Vvm
With this equation, because the common subscript is on the outside, the resultant subscripts are presented backwardly. Of course, you can switch the order of the terms to make the common subscript be on the inside and get the same answer, but knowledge of these tricks either way will be very helpful.
KINEMATICS POSTMORTEM
We have now come to the end of Non-Rotational Kinematics (at least, what I will consider 'Non-Rotational Kinematics' in my notes), our studies in Vector Addition, Projectile Motion, Relative Motion, and everything else. Let us review the similarities between the vector decompositions of the various motions: The core principle behind decomposition is that when a vector is not directly in the x or y direction, break the vector in its components. Why do we do this?
Vector Addition: We create components in order to make a right triangle so we can use the Pythagorean Theorem and the Trigonometric functions.
Projectile Motion: We resolve the initial velocity into components because we use different equations in the X and Y directions for projectile motion. We use the equation for constant velocity in the X direction (because acceleration is zero), and in the Y direction we use the U.A.M. equations for free fall.
Relative Motion: Same as Vector Addition, decompose to make a right triangle.
For ever and ever, you will use vectors and vector decomposition in your daily life, because if you are not living as a physicist, are you truly living?
