Calculus Based Physics Formulas: Mechanics

This is just a basic equation list, explanations can be found elsewhere. For the most part derivations are done for you, but it is beneficial to understand how an equation goes from one form to another.

One dimensional Equations of motion (along a single vector direction)
Velocity as a function of time : v_{xf} = v_{xi} + a_x t
Position as a function of time:  x_f = x_i + v_{xi}t + \frac{1}{2}  a_x t^2
Velocity as a function of position: v^2_{xf} = v^2_{xi} + 2a_x ( x_f &#8211; x_i)<br />

Projectile Motion
Horizontal motion
Velocity along x: v_{xi} = v_i cos(\theta)
Position from position as a function of time: x_f = v_i cos(\theta)t
Max Horizontal dist: R = v^2_i \frac{sin( 2 \theta_i)}{g}

Vertical Motion
Velocity along y:v_{yi} = v_i sin(\theta)
Position: from position as a function of timey_f = y_i + v_{yi}t &#8211; \frac{1}{2} g*t^2
Maximum Height:  h_{max} = v^2_{i} \frac{sin(\theta_i)}{2g}

Circular Motion
Radial Acc: a_r = v^2_r = a cos( \theta)
Tan. Acc:a_t = \frac{d \mid \vec {v}\mid}{dt} = a sin(\theta)= r \alpha
Total Acc (magnitude) from Pythagoras: a = \sqrt{a_r^2 + a_t^2}

The Laws of Motion
Newtons Second Law: \sum{F_{x,y, or z}} = ma_{x,y, or z}
Equilibrium Conditions: \sum {F_{x,y, or z}} =  0
Force of Static Friction F_{s max} = \mu_s*n
Force from Kinetic Friction F_{k max} = \mu_k*n

Force/Work
Constant Force:  w_{net} = \vec{f_{net}}*\delta r = F * r cos(\theta) = \delta K
Variable Force: w_{net} = \int f_{net} d \vec{r}
Hooke’s Law: f_s = -k x
Spring Work: w = \frac{1}{2} k x_i^2  &#8211;  \frac{1}{2} k x^2_f
Kinetic energy: k = \frac{1}{2} m v^2
Work – kinetic energy theorem: w_{net} = \delta k = k_f &#8211; k_i, k_f = k_i
Power: P = \frac {\Delta w}{\Delta t} ,p = \frac {de}{dt} , P = \vec{f} \vec{v}
gravitational potential energy: U = mgh
conservation of mechanical energy: E = K_f + U_f = K_i + U_i = const. + \mid f_k \delta x \mid
elastic collision conserved moment and KE: v_{1f} = (\frac{m_1 &#8211; m_2}{m_1 + m_2}) v_{1i} + (\frac {2 m_2}{m_1 + m_2}) v_{2i}
2d elastic (comp):  m_1 v_{1ix} + m_2 v_{2ix} =  m_1 v_{1fx} + m_2 v_{2fx},m_1 v_{1ix}  =  m_1 v_{1f} cos(\theta) + m_2 v_{2fx}cos(\phi)
KE conservation for elastic:  \frac{1}{2} m_i v_i^2 +\frac{1}{2} m_{2i} v_{2i}^2 =  \frac{1}{2} m_1 i v_{1f} i^2 +  \frac{1}{2} m_2 i v_{2f} i^2
Momentum: \vec{P} = m \vec{{v}

Mass
Center of mass (comp): x_{cm} = \frac{\sum_{i=1}^{n} m_i x_i  }{m}
Position vector for CM: \vec{r_{cm}} = x_{cm} \vec {i} + y_{cm} \vec {j} + z_{cm} \vec {k}
Continuous mas dist: x_{cm} =\frac {1}{m}  \int{\lambda dx}
Mass of Uniform: m= \int{\lambda dx}
Linear Mass Dist: \lambda = \frac{m}{l} = \frac{dm}{dl}
Area Mass Dist: \omega = \frac{m}{a} = \frac{dm}{da}

Rotational Motion
angular speed: \omega = \frac{d\theta}{dt}
angular acceleration: \alpha = \frac{d\omega}{dt} ,\frac{a_t}{r},\frac{\tau}{I}
Moment of Inertia: I = m_i r_i^2,I = \int r^2 dm,I = \int (density)r^2 dv,I = I_cm + mD^2
Rotational KE: K_R = \frac{1}{2} I \omega^2
Work: \frac{1}{2} I \omega_f^2 &#8211; \frac{1}{2}  I \omega_i^2
Net torque : \sum\tau = I \alpha, \sum \tau =\frac {dL}{dt}
Work : W = \int_{\theta_f}^{\theta_i} t  d\theta
Power : P = \tau \omega
Angular Momentum : L = I \omega
Torque: : \tau = rF sin(\theta)

Moments of Inertia
Hoop : I_{cm} =mr^2
Cylinder (hollow) : I_{cm} = \frac{1} {2} m(r_1^2 + r_2^2)
Cylinder : I_{cm} = \frac{1} {2} mr^2
Rectangular Plate : I_{cm} = \frac{1} {12} m(a^2 + b^2)
Rod (center rotate):  I_{cm} =\frac{1} {12} mL^2
Rod (end Rotate):  I_{cm} =\frac{1} {3} mL^2
Solid Sphere : I_{cm} = \frac{2} {5} mr^2
Spherical Shell : I_{cm} = \frac{2} {3} mr^2

Units in Physics (mechanical, electricity, magnetism, light and optics) including Si units.

This is a reference list with notes of all SI and derived units in physics. The notes provide a brief explanation of some of the more confusing elements, but be warned that the full explanation could take many pages, and may be explained elsewhere on this website.

Physics has only 5 base units. (Plus the SI units Mole and Candela, but these are rarely used in Physics.)

Name Abbreviation (Symbol) Standard Unit Notes
Name Abbreviation (Symbol) Standard Unit Notes
Length l, x (for distances) Meter (m) A meter is defined as the distance light travels in a vacumm in \frac{1}{299 792 458} of a second (in physics it is customary to use metric measurements although the basic principles apply if you to use feet instead of meters)
Mass m, M (when used with measurements in meters) Kilogram (kg) A kilogram is defined as the weight of a specific platinum-iridium cylinder
Time t Second (s) Seconds are defined as 9,192,631,770 vibrations of radiation from a cesium atom
Temparature T Kelvin (K) A degree kelvin is defined as \frac{1}{273.16} of the distance between absolute 0 and the triple point of water
Electric Current I Ampere (A) An ampere is the amount of charge (C) passing through a surface per second, and is defined as the current which produces a force of 2*10^{-7} newtons per meter of length between two infinitely long, perfectly straight and parallel conductors with an infinitely small cross section separated by one meter in a vacuum..

Each of these base units is defined on fundamental constants, and all other units are based on these five units. At times it useful to break longer equations down to their most basic units to determine if the equation makes sense. The most common combinations of these basic units are given their own symbols and names. These common units are as follows.
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Using the Rules of Physics to Find the Mass of the Planet Crouching Tiger, Hidden Dragon Takes Place on.

The clip that is referenced is near the end of the movie after the two female characters fight at the school, and runs for about 2 -3 minutes.

It doesn’t take any physics experience to realize that “Crouching Tiger Hidden Dragon” is a bad physics movie, however with physics we can show just how bad it really is. The clip I selected runs only a few seconds long but portrays many of the formula’s that one would use in a simple mechanics physics course. Through careful calculations and a little estimation (all double checked) I was able to determine a range of details including the acceleration of gravity on the “planet” this movie takes place in, the diameter of the planet, and the density of the planet. I was also able to determine a few specific examples aside from the acceleration of gravity that show that this movie is a BPM.

To calculate the acceleration of gravity I used two different scenes, the first scene used was from the beginning of the clip where the actress jumped a single full jump over part of a body of water. By finding the actresses real height I estimated the following ratio of 1.22 meters = 1cm (second trial 1cm = 1.5m). I then calculated the ground distance of her jump to be about 6 m (second trial 9m) long and her angle of jump to be 50 degrees, using these two pieces of information i found the actual distance she covered in her arc to be 9.48 meters (second trial 14m). The time of this jump is roughly 3 seconds long. Using a grapher on my calculator I graphed a curve of her jump (trial one) that started at (0,0) reached a maximum of (1.5, 1.4) and (3,0) the equation of this curve (basic height function) is: F(x) = .8124x^2 +2.437x , where x is the time and F(x) is the distance above the ground. I then found the second derivative of the function for an instantaneous acceleration of 1.6248 m/s^2, however actual distances predict she should have been traveling at 2.5 m/s^2. Because all the numbers were estimated it is easy to see how a little bit can make a big difference, in this case by increasing trial ones height to 2m and decreasing the time the jump takes to 2.6 seconds (both of which are easily in the margin of error) we get an acceleration of 2.45 m/s^2. This whole process was then repeated with a larger scale and ended with an instantaneous acceleration of 2.5m/s^2 confirming the earlier number because the distances are roughly double.

Obviously this is a BPM because acceleration of a falling object is 9.8 m/s^2, had she been falling at this speed she would have traversed 11m instead of just under 9.48m.

This same result is seen in the second example scene where she “falls”� onto the pond. Again using her height as the base we find she fell about 12m down in three seconds, which when plugged into the distance formula gives us the acceleration we were expecting from the above (2.5 m/s^2).
The third example is using the gravity constant and the above acceleration to find the mass of the planet and the distance from the center. Starting with the formula: 2.5 = 6.67×10^-11 (M/d^2) where 2.5 is the acceleration and M is the planets weight in kilograms. Solving the equation for “d” we find that d = .000005*m1/2 (Solving for M gives us 3.7 *1010*d^2) unfortunately we only have a single equation so it is impossible to solve for a single number M in the prior equation, however we can use another M as a base and then solve.

The “D” we have is an exponential function that gives the Distance from the center of a planet with mass of “M”. Therefore we now have a formula that can calculate the information of any planet with mass M (or distance from the same) that has an acceleration from gravity of 2.5 m/s^2.
Volume = .75*pi*(.000005*m^½)^3
Density = M/ .75*pi*(.000005*m^½ )^3
Circumference = 2*pi* (.000005*m^ ½ )
For example:
To if we use earth as the mass we find that we are 1.2*107m away from earth ( this number puts us outside the moon’s orbit), that the volume of the space closer to earth is 4.3*10^21 m^3, the density is 1386.1 kg /m and the circumference of the space is 7.6*107m.
Comparing those numbers with the real ones:
Distance from the center on the surface of the earth: 6378.1 km
Earths volume: 1.0832�?1012 km^3
Density: 5,515 kg/m³
Circumference: 40,075. km
We find that to have the acceleration of earths gravity at 2.5 m/s^2 the actors would have to have been 1881.44 times as far away from the center of earth than if they had had 9.8 m/s^2 as their acceleration. Using the rule of inverse squares the actress with mass 42kg has a weight of 411.6n on earth but would only weigh 1/3545689th (.000116n) as much as if she was on earth.
Of course there are many many other examples in the same scene that can be used, but they all focus on the same things.