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These are the solutions to the Mathematics/Physics questions on Mechanics.
This consists of topics in Statics and Dynamics.
When applicable, the TI-84 Plus CE calculator (also applicable to TI-84 Plus calculator) solutions are provided
for some questions.
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(1.) Basic Formulas
$
W = mg \\[3ex]
where \\[3ex]
W = weight\;(N) \\[3ex]
g = \text{acceleration due to gravity}\; (ms^{-2}) \\[3ex]
m = \text{mass}\;(kg) \\[3ex]
$
(2.) Basic Equations of Motion
$
(a.)\;\; v = u + at \\[3ex]
(b.)\;\; s = ut + \dfrac{1}{2}at^2 \\[5ex]
(c.)\;\; v^2 = u^2 + 2as \\[5ex]
\underline{\text{Working towards gravity}} \\[3ex]
a = g \\[5ex]
\underline{\text{Working aganist gravity}} \\[3ex]
a = -g \\[5ex]
$
where:
u = initial velocity
v = final velocity
a = acceleration
t = time
s = distance
g = acceleration due to gravity
(3.) Total Momentum
$
m_1 = \text{mass of first body} \\[4ex]
v_1 = \text{velocity of first body} \\[4ex]
m_2 = \text{mass of second body} \\[4ex]
v_2 = \text{velocity of second body} \\[5ex]
\underline{\text{Before Collision: Moving in Same Direction}} \\[4ex]
\text{momentum of first body} = m_1v_1 \\[4ex]
\text{momentum of second body} = m_2v_2 \\[4ex]
\text{total momentum} = m_1v_1 + m_2v_2 \\[5ex]
\underline{\text{After Collision: If Stuck Together}} \\[3ex]
\text{mass of: first body and second body} = m_1 + m_2 \\[4ex]
\text{common velocity} = v \\[3ex]
\text{total momentum} = v(m_1 + m_2) \\[5ex]
\underline{\text{Based on the Principle of Conservation of Momentum}} \\[3ex]
\text{total momentum before collision = total momentum after collision} \\[3ex]
m_1v_1 + m_2v_2 = v(m_1 + m_2) \\[5ex]
\underline{\text{Before Collision: Moving in Opposite Directions}} \\[4ex]
\text{momentum of first body} = m_1v_1 \\[4ex]
\text{momentum of second body} = m_2 * -v_2 = -m_2v_2 \\[4ex]
\text{The negative value of the velocity is because the second body is moving in opposite direction} \\[3ex]
\text{total momentum} = m_1v_1 - m_2v_2 \\[5ex]
\underline{\text{After Collision: If Stuck Together}} \\[3ex]
\text{mass of: first body and second body} = m_1 + m_2 \\[4ex]
\text{common velocity} = v \\[3ex]
\text{total momentum} = v(m_1 + m_2) \\[5ex]
\underline{\text{Based on the Principle of Conservation of Momentum}} \\[3ex]
\text{total momentum before collision = total momentum after collision} \\[3ex]
m_1v_1 - m_2v_2 = v(m_1 + m_2) \\[5ex]
$
(4.) Forces acting on a body
$
F_x = F \cos\theta \\[3ex]
F_y = F \sin\theta \\[3ex]
$
where:
$F$ = magnitude of the force
$F_x$ = horizontal component (x-component) of the force
$F_y$ = vertical component (y-component) of the force
θ = angle the force makes with the positive x-axis (measured counterclockwise).
$
F = \sqrt{F_x^2 + F_y^2} \\[3ex]
\theta = \tan^{-1}\left(\dfrac{F_y}{F_x}\right) \\[5ex]
$
(5.) A Body in Equilibrium
$
\Sigma F_x = 0 \\[3ex]
\Sigma F_y = 0 \\[5ex]
$
(6.) Gravitational Force
$
\underline{\text{Newton's Law of Universal Gravitation}} \\[3ex]
F = \dfrac{Gm_1m_2}{r^2} \\[5ex]
\text{where} \\[3ex]
\text{F is the gravitational force between two bodies} \\[3ex]
\text{G is the gravitational constant} \\[3ex]
m_1 \text{ is the mass of the first body} \\[3ex]
m_2 \text{ is the mass of the second body} \\[3ex]
\text{r is the distance between the centers of mass of the two bodies.} \\[5ex]
\underline{\text{Escape Velocity}} \\[3ex]
v_e = \sqrt{\dfrac{2GM}{R}} \\[5ex]
v_e = \sqrt{2gR} \\[3ex]
g = \dfrac{GM}{R^2} \\[5ex]
\text{where} \\[3ex]
v_e \text{ is the escape velocity} \\[3ex]
\text{G is the gravitational constant} \\[3ex]
\text{M is the mass of the celestial body} \\[3ex]
\text{R is the radius of the celestial body} \\[3ex]
\text{g is acceleration due to gravity on the surface of the celestial body} \\[5ex]
$