Useful Things

NASAPhysics

Online Course

Useful Things

Eqn #	Eqn	Units	Calculates
1.1	v = d/t	m/s	velocity
1.2	t = d/v	s	time
1.3	a = v/t	m/s²	acceleration
1.4	v = at	m/s	velocity
1.5	v_f = v_i + at	m/s	final velocity
1.6	v_ave = (v_i + v_f) / 2	m/s	average velocity
1.7	d = ½ at²	m	distance
1.8	v = gt	m/s	velocity
1.9	d = ½ gt²	m	distance
1.10	F = (Gm₁m₂)/r²	newtons	gravitational force
1.11	x = v_ixt + ½ a_xt²	m	x position
1.12	v_fx = v_ix +a_xt	m/s	final x velocity
1.13	y = v_iyt + ½a_yt²	m	Y position
1.14	v_fy = v_iy +gt	m/s	final y velocity
1.15	p² = a³	yr, A.U.	Kepler’s 3^rd Law
1.16	v = (GM/r)^½	km/s	orbital velocity
1.17	v = (2GM/r)^½	km/s	escape velocity
1.18	F = ma	newtons	force to accelerate a mass
2.1	TE = PE + KE + …	J, joules	total energy
2.2	KE = ½ mv²	J, joules	kinetic energy
2.3	PE = mgh	J, joules	potential energy
2.4	PE₁ + KE₁ = PE₂ + KE₂	J, joules	conservation of energy
2.5	v² = 2g(H-y)	m/s	roller coaster velocity
2.6	F_c = mv²/r	newtons	centripetal force
2.7	h ≥ 2.5r	m	height for roller coaster to safely loop-d-loop
2.8	p = mv	kg m/s	momentum
2.9	p = Ft	kg m/s	momentum
2.10	L = mvr	kg • m²/s	angular momentum
3.1	C = (F-32)*5/9	degrees	Centigrade to Fahrenheit
3.2	F = (C*9/5) + 32	degrees	Fahrenheit to Centigrade
3.3	C = K -273	degrees	Kelvin to Centigrade
3.4	K = C +273	none	Centigrade to Kelvin
3.5	F = (K-273)*9/5 + 32	degrees	Kelvin to Fahrenheit
3.6	K = (F-32)*5/9 + 273	none	Fahrenheit to Kelvin
3.7	ΔL = α L ΔT	mm	change in length heated bar
3.8	Q = c m ΔT	calories	heat
3.9	Q = mH_f	calories	heat to melt solid
3.10	Q = mH_v	calories	heat to vaporize liquid
3.11	H = (k t A T)/d	calories	conductive heat
4.1	P = F/A	pascals	fluid pressure
4.2	P = ρgh	pascals	pressure with depth
4.3	F_b = ρ_fVg	newtons	buoyant force
4.4	PV = constant	pascals, m³	Boyle’s Law
4.5	V = constant * T	m³	Charles/Gay-Lussac’s Law
4.6	PV = nRT	pascals, m³	Ideal Gas Law
4.7	P ∝ T	pascals, °C	pressure proportional to temperature
4.8	P ∝1 / V	pascals, m³	pressure inversely proportional to volume
5.1	i = r	degrees	Law of Reflection
5.2	n = c/v	none	index of refraction
5.3	sin i/sin r = v₁/v₂= n₂/n₁	none	Snell’s Law
6.1	f = c/λ	hertz	frequency of light
7.1	T = 2,897,000/ λ_max	K	temperature of a star
7.2	E = hf	joules	energy of EM wave
7.3	E = hc/ λ	joules	energy as function of wavelength
7.4	t = t_o / sqrt (1-(v²/c²))	s	time in two reference frames
7.5	ɣ = 1/(sqrt(1-(v²/c²))	none	gamma
7.6	t = ɣt_o	s	time in two reference frames

Letters Used in Equations

A.U. = astronomical Unit = 1.5 x 10⁸ km

A = area

a = acceleration

a = semi-major axis of orbiting body

c = speed of light = 3 x 10⁸ m/s

c = specific heat

d = distance, thickness

EM = electromagnetic

EMS = electromagnetic spectrum

F = force

f = frequency of EM radiation

Fc = centripetal force

Fg = gravitational force

G = universal gravitational constant = 6.67 x 10^-11

Nm²/kg²

g = acceleration of gravity = 9.8 m/s²

H = maximum height of roller coaster

H = heat flow

Hf = heat of fusion (melting)

Hv = heat of vaporization (evaporating)

h = height

h = Planck Constant = 6.636 x 10^-34 Js

i = angle of incidence

k = thermal conductivity

KE = kinetic energy

ΔL = change in length

L = angular momentum

PE = potential energy

M, m = mass

n = number of moles of a substances (n x 6.022 x

10²³ molecules)

n = refractive index

P = pressure

p = period of an orbiting body

p = momentum

Q = heat

R = the universal gas constant = 8.31 J/molK

r = distance, radius

r = angle of reflection

TE = total energy

T = temperature

ΔT = change in temperature

t = time

t_o = time in alternative reference frame

V = volume

v = velocity

v_f = final velocity

v_fx = final velocity in x direction

v_fy = final velocity in y direction

v_i = initial velocity

v_ix = initial velocity in x direction

v_iy = initial velocity in y direction

y = instantaneous height of roller coaster

yr = year

Greek Letters

ɣ (gamma) = factor in special relativity

λ (lambda) = wavelength

λ_max = wavelength of max emission

ρ (rho) = density

ρ_f =fluid density

Units Conversion factors

Distance: nm = nanometer = 1 x 10^-9 m

Energy: calorie = 1cal = 4.18 J

Energy: joule = 1 J = kg*m²/ s²

Force: newton = 1 N = kg*m/s²

Frequency: hertz = 1 Hz = 1/s

Pressure: pascal = 1 Pa = 1 N/m²

Scientific Notation

Signigicant Figures

Any number can be written as the product of two numbers, an integer and 10 to some power. This is especially useful for very small and very large numbers, such as

0.00000043 = 4.3 x 10^-7

299,873,000 = 2.99873 x 10⁸

1 = 10⁰

10 = 10¹

100 = 10²

1,000 = 10³

1,000,000 = 10⁶(a million)

1,000,000,000 = 10⁹ (a billion)

0.1 = 10^-1

0.01 = 10^-2

0.001 = 10^-3

0.000001 = 10^-6 (a millionth)

0.000000001 = 10^-9 (a billionth)

To multiply using scientific notation multiple the integer terms and add the exponents of the 10^x terms.

(2.3 x 10³) x (1.5 x 10⁴) = 3.4 x 10⁷

To divide using scientific notation divide the integer terms and subtract the exponents of the 10^x terms.

(5 x 10⁵) / (2 x 10³) = 2.5 x 10²

To raise to a power using scientific notation raise the integer terms to the indicated power and multiple the exponent of the 10^x terms by the power.

(3 x 10³)⁴ = (3 x 3 x 3 x 3 x10³*⁴) = 81 x 10¹² = 8.1 x 10¹³

To find the xth root of a number using scientific notation, find the root of the integer term and divide the exponent of the 10^x terms by the power.

(27 x 10⁹)^1/3= 3 x 10³

(note that a fractional exponent such as ^1/3 means to take the 3^rd root).

Have you ever had a problem such as this:

X = 292/2.1

and turned in an answer like this:

X = 139.047619

If you look at the two numbers in the original problem, 292 and 2.1, each has only two or three digits. We have no information that the numbers aren’t really:

292.000000 or 292.439 or 291.56238

and 2.10000000, or 2.11, or 2.09876542345.

In other words we don’t know how accurately known each number really is. We say that 292 has 3 significant digits, and 2.1 has two significant digits.

Scientists have agreed that answers should only be given to the same number of significant digits as the least well-known number in any operation. So for this problem there are only two digits in 2.1 so the answer can only have two significant digits:

X = 140

Notice that writing 139 implies that we know the last digit is a 9, and not an 8 or a 0. We don’t know that, so we round up to 140, with the 0 at the end not counting as a significant digit.

Every time you do a calculation look to see how many digits there are in the number with the least significant digits, and make your answer match that precision.