A coin is dropped from a hot air-balloon that is \(250 \, \text{m}\) above the ground rising at \(11 \, \text{m/s}\) upwards. For the coin, assume up is positive and find the following:(a) The maximum height reached (in \(\text{meters}\)). , (b) The position, \(4.20 \, \text{seconds}\) after being released. , (c) The velocity \(4.20 \, \text{s}\) after being released. , (d) The time (in seconds) before it hits the ground :

A coin is dropped from a hot air-balloon that is \(250 \, \text{m}\) above the ground rising at \(11 \, \text{m/s}\) upwards. For the coin, assume up is positive and find the following: • Nerd Notes

1. Maximum Height Reached by the Coin

Step	Formula Derivation	Reasoning
1	\(v^2 = u^2 + 2as\)	Kinematic equation for motion, with \(v\) as final velocity, \(u\) as initial velocity, \(a\) as acceleration, and \(s\) as displacement.
2	\(0 = (11, \text{m/s})^2 – 2 \times 9.81, \text{m/s}^2 \times s\)	At maximum height, final velocity \(v = 0\), initial velocity \(u = 11, \text{m/s}\) upward, acceleration \(a = -9.81, \text{m/s}^2\) (gravity acts downward).
3	Solve for \(s\)	Calculate the displacement \(s\).

2. Position 4.20 Seconds After Being Released

Step	Formula Derivation	Reasoning
1	\(s = ut + \frac{1}{2}at^2\)	Kinematic equation for displacement.
2	\(s = 11, \text{m/s} \times 4.20, \text{s} – \frac{1}{2} \times 9.81, \text{m/s}^2 \times (4.20, \text{s})^2\)	Substitute values for \(u\), \(a\), and \(t\) (time after release).

3. Velocity 4.20 Seconds After Being Released

Step	Formula Derivation	Reasoning
1	\(v = u + at\)	Kinematic equation for velocity.
2	\(v = 11, \text{m/s} – 9.81, \text{m/s}^2 \times 4.20, \text{s}\)	Substitute values for \(u\), \(a\), and \(t\).

4. Time Before Coin Hits the Ground

Step	Formula Derivation	Reasoning
1	\(s = ut + \frac{1}{2}at^2\)	Kinematic equation for displacement.
2	\(-250, \text{m} = 11, \text{m/s} \times t – \frac{1}{2} \times 9.81, \text{m/s}^2 \times t^2\)	Substitute values for \(s\) (downward displacement), \(u\), and \(a\). Solve for \(t\).

Let’s perform the calculations for each part.

The results for each part are as follows:

Maximum height reached by the coin: \(\boxed{256.17, \text{meters}}\)
Position of the coin 4.20 seconds after being released: \(\boxed{209.68, \text{meters}}\) above the ground
Velocity of the coin 4.20 seconds after being released: \(\boxed{-30.20, \text{m/s}}\) (downward, as indicated by the negative sign)
Time before the coin hits the ground: \(\boxed{8.35, \text{seconds}}\)

Kinematics	Forces
\(\Delta x = v_i t + \frac{1}{2} at^2\)	\(F = ma\)
\(v = v_i + at\)	\(F_g = \frac{G m_1 m_2}{r^2}\)
\(v^2 = v_i^2 + 2a \Delta x\)	\(f = \mu N\)
\(\Delta x = \frac{v_i + v}{2} t\)	\(F_s =-kx\)
\(v^2 = v_f^2 \,-\, 2a \Delta x\)

Circular Motion	Energy
\(F_c = \frac{mv^2}{r}\)	\(KE = \frac{1}{2} mv^2\)
\(a_c = \frac{v^2}{r}\)	\(PE = mgh\)
\(T = 2\pi \sqrt{\frac{r}{g}}\)	\(KE_i + PE_i = KE_f + PE_f\)
	\(W = Fd \cos\theta\)

Momentum	Torque and Rotations
\(p = mv\)	\(\tau = r \cdot F \cdot \sin(\theta)\)
\(J = \Delta p\)	\(I = \sum mr^2\)
\(p_i = p_f\)	\(L = I \cdot \omega\)

Simple Harmonic Motion	Fluids
\(F = -kx\)	\(P = \frac{F}{A}\)
\(T = 2\pi \sqrt{\frac{l}{g}}\)	\(P_{\text{total}} = P_{\text{atm}} + \rho gh\)
\(T = 2\pi \sqrt{\frac{m}{k}}\)	\(Q = Av\)
\(x(t) = A \cos(\omega t + \phi)\)	\(F_b = \rho V g\)
\(a = -\omega^2 x\)	\(A_1v_1 = A_2v_2\)

Constant	Description
[katex]g[/katex]	Acceleration due to gravity, typically [katex]9.8 , \text{m/s}^2[/katex] on Earth’s surface
[katex]G[/katex]	Universal Gravitational Constant, [katex]6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2[/katex]
[katex]\mu_k[/katex] and [katex]\mu_s[/katex]	Coefficients of kinetic ([katex]\mu_k[/katex]) and static ([katex]\mu_s[/katex]) friction, dimensionless. Static friction ([katex]\mu_s[/katex]) is usually greater than kinetic friction ([katex]\mu_k[/katex]) as it resists the start of motion.
[katex]k[/katex]	Spring constant, in [katex]\text{N/m}[/katex]
[katex] M_E = 5.972 \times 10^{24} , \text{kg} [/katex]	Mass of the Earth
[katex] M_M = 7.348 \times 10^{22} , \text{kg} [/katex]	Mass of the Moon
[katex] M_M = 1.989 \times 10^{30} , \text{kg} [/katex]	Mass of the Sun

Variable	SI Unit
[katex]s[/katex] (Displacement)	[katex]\text{meters (m)}[/katex]
[katex]v[/katex] (Velocity)	[katex]\text{meters per second (m/s)}[/katex]
[katex]a[/katex] (Acceleration)	[katex]\text{meters per second squared (m/s}^2\text{)}[/katex]
[katex]t[/katex] (Time)	[katex]\text{seconds (s)}[/katex]
[katex]m[/katex] (Mass)	[katex]\text{kilograms (kg)}[/katex]

Variable	Derived SI Unit
[katex]F[/katex] (Force)	[katex]\text{newtons (N)}[/katex]
[katex]E[/katex], [katex]PE[/katex], [katex]KE[/katex] (Energy, Potential Energy, Kinetic Energy)	[katex]\text{joules (J)}[/katex]
[katex]P[/katex] (Power)	[katex]\text{watts (W)}[/katex]
[katex]p[/katex] (Momentum)	[katex]\text{kilogram meters per second (kgm/s)}[/katex]
[katex]\omega[/katex] (Angular Velocity)	[katex]\text{radians per second (rad/s)}[/katex]
[katex]\tau[/katex] (Torque)	[katex]\text{newton meters (Nm)}[/katex]
[katex]I[/katex] (Moment of Inertia)	[katex]\text{kilogram meter squared (kgm}^2\text{)}[/katex]
[katex]f[/katex] (Frequency)	[katex]\text{hertz (Hz)}[/katex]

Prefix	Symbol	Power of Ten
Pico-	p	[katex]10^{-12}[/katex]
Nano-	n	[katex]10^{-9}[/katex]
Micro-	µ	[katex]10^{-6}[/katex]
Milli-	m	[katex]10^{-3}[/katex]
Centi-	c	[katex]10^{-2}[/katex]
Deci-	d	[katex]10^{-1}[/katex]
(Base unit)	–	[katex]10^{0}[/katex]
Deca- or Deka-	da	[katex]10^{1}[/katex]
Hecto-	h	[katex]10^{2}[/katex]
Kilo-	k	[katex]10^{3}[/katex]
Mega-	M	[katex]10^{6}[/katex]
Giga-	G	[katex]10^{9}[/katex]
Tera-	T	[katex]10^{12}[/katex]

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