# You stand at the edge of a vertical cliff and throws a stone vertically upwards. The stone leaves your hand with a speed v = 8.0 m/s. The time between the stone leaving your hand and hitting the sea is 3.0 s. Assume air resistance is negligible. Calculate:

1. (a) The maximum height reached by the stone (the distance above the cliff). (3 points)
2. (b) The time taken by the stone to reach its maximum height. (2 points)
3. (c) The height of the cliff. (2 points)
1. 3.26 m above the cliff
2. .82 s
3. 20.15 m
0

1. Maximum Height Reached by the Stone

Step Formula Derivation Reasoning
1 v^2 = u^2 + 2as Kinematic equation, with v as final velocity (0 at max height), u as initial velocity, a as acceleration (gravity), and s as displacement.
2 0 = (8.0, \text{m/s})^2 – 2 \times 9.81, \text{m/s}^2 \times s_{\text{max}} At max height, final velocity is 0.
3 Solve for s_{\text{max}} Maximum height above the cliff.

2. Time to Reach Maximum Height

Step Formula Derivation Reasoning
1 v = u + at Kinematic equation for velocity.
2 0 = 8.0, \text{m/s} – 9.81, \text{m/s}^2 \times t_{\text{max}} Final velocity is 0 at max height.
3 Solve for t_{\text{max}} Time to reach max height.

3. Height of the Cliff

Total Time from Throw to Sea

Step Formula Derivation Reasoning
1 s = ut + \frac{1}{2}at^2 Displacement formula for the entire journey, where s is total displacement (height of the cliff), u is initial velocity, t is total time, and a is acceleration (gravity). Note gravity is negative in this case.
2 s_{\text{cliff}} = 8.0, \text{m/s} \times 3.0, \text{s} – \frac{1}{2} \times 9.81, \text{m/s}^2 \times (3.0, \text{s})^2 Substituting the values for u, t, and g.
3 Solve for s_{\text{cliff}} Height of the cliff.

Performing the calculations above yield the following results:

1. Maximum height reached by the stone above the cliff: \boxed{3.26, \text{m}}
2. Time taken by the stone to reach its maximum height: \boxed{0.82, \text{s}}
3. Height of the cliff: \boxed{20.15, \text{m}}

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1. 3.26 m above the cliff
2. .82 s
3. 20.15 m

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KinematicsForces
\Delta x = v_i t + \frac{1}{2} at^2F = ma
v = v_i + atF_g = \frac{G m_1m_2}{r^2}
a = \frac{\Delta v}{\Delta t}f = \mu N
R = \frac{v_i^2 \sin(2\theta)}{g}
Circular MotionEnergy
F_c = \frac{mv^2}{r}KE = \frac{1}{2} mv^2
a_c = \frac{v^2}{r}PE = mgh
KE_i + PE_i = KE_f + PE_f
MomentumTorque and Rotations
p = m v\tau = r \cdot F \cdot \sin(\theta)
J = \Delta pI = \sum mr^2
p_i = p_fL = I \cdot \omega
Simple Harmonic Motion
F = -k x
T = 2\pi \sqrt{\frac{l}{g}}
T = 2\pi \sqrt{\frac{m}{k}}
ConstantDescription
gAcceleration due to gravity, typically 9.8 , \text{m/s}^2 on Earth’s surface
GUniversal Gravitational Constant, 6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2
\mu_k and \mu_sCoefficients of kinetic (\mu_k) and static (\mu_s) friction, dimensionless. Static friction (\mu_s) is usually greater than kinetic friction (\mu_k) as it resists the start of motion.
kSpring constant, in \text{N/m}
M_E = 5.972 \times 10^{24} , \text{kg} Mass of the Earth
M_M = 7.348 \times 10^{22} , \text{kg} Mass of the Moon
M_M = 1.989 \times 10^{30} , \text{kg} Mass of the Sun
VariableSI Unit
s (Displacement)\text{meters (m)}
v (Velocity)\text{meters per second (m/s)}
a (Acceleration)\text{meters per second squared (m/s}^2\text{)}
t (Time)\text{seconds (s)}
m (Mass)\text{kilograms (kg)}
VariableDerived SI Unit
F (Force)\text{newtons (N)}
E, PE, KE (Energy, Potential Energy, Kinetic Energy)\text{joules (J)}
P (Power)\text{watts (W)}
p (Momentum)\text{kilogram meters per second (kgm/s)}
\tau (Torque)\text{newton meters (Nm)}
I (Moment of Inertia)\text{kilogram meter squared (kgm}^2\text{)}
f (Frequency)\text{hertz (Hz)}

General Metric Conversion Chart

Example of using unit analysis: Convert 5 kilometers to millimeters.

1. Start with the given measurement: \text{5 km}

2. Use the conversion factors for kilometers to meters and meters to millimeters: \text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}

3. Perform the multiplication: \text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}} = 5 \times 10^3 \times 10^3 \, \text{mm}

4. Simplify to get the final answer: \boxed{5 \times 10^6 \, \text{mm}}

Prefix

Symbol

Power of Ten

Equivalent

Pico-

p

10^{-12}

Nano-

n

10^{-9}

Micro-

µ

10^{-6}

Milli-

m

10^{-3}

Centi-

c

10^{-2}

Deci-

d

10^{-1}

(Base unit)

10^{0}

Deca- or Deka-

da

10^{1}

Hecto-

h

10^{2}

Kilo-

k

10^{3}

Mega-

M

10^{6}

Giga-

G

10^{9}

Tera-

T

10^{12}

1. Some answers may be slightly off by 1% depending on rounding, etc.
2. Answers will use different values of gravity. Some answers use 9.81 m/s2, and other 10 m/s2 for calculations.
3. Variables are sometimes written differently from class to class. For example, sometime initial velocity v_i is written as u ; sometimes \Delta x is written as s .
4. Bookmark questions that you can’t solve so you can come back to them later.
5. Always get help if you can’t figure out a problem. The sooner you can get it cleared up the better chances of you not getting it wrong on a test!

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