 # 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: The maximum height reached by the stone (the distance above the cliff) The time taken by the stone to reach its maximum height. The height of the cliff.

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

## Written By Prof Phy AI

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 ## Suggest an Edit

##### Nerd-Notes.com
KinematicsForces
\Delta x = v_i \cdot t + \frac{1}{2} a \cdot t^2F = m \cdot a
v = v_i + a \cdot tF_g = \frac{G \cdot m_1 \cdot m_2}{r^2}
a = \frac{\Delta v}{\Delta t}f = \mu \cdot N
R = \frac{v_i^2 \cdot \sin(2\theta)}{g}
Circular MotionEnergy
F_c = \frac{m \cdot v^2}{r}KE = \frac{1}{2} m \cdot v^2
a_c = \frac{v^2}{r}PE = m \cdot g \cdot h
KE_i + PE_i = KE_f + PE_f
MomentumTorque and Rotations
p = m \cdot v\tau = r \cdot F \cdot \sin(\theta)
J = \Delta pI = \sum m \cdot r^2
p_i = p_fL = I \cdot \omega
Simple Harmonic Motion
F = -k \cdot 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}
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 (kg·m/s)}
\tau (Torque)\text{newton meters (N·m)}
I (Moment of Inertia)\text{kilogram meter squared (kg·m}^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!