AP Physics

Unit 1 - Vectors and Kinematics

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# (a) Rock’s Speed Just Before It Falls Into the River

Step Derivation/Formula Reasoning
1 v_i = 20 \, \text{m/s} Initial velocity of the rock thrown upward.
2 h = -42 \, \text{m} Final displacement of the rock from the point of throw to the river is -42 meters (negative since it’s downward).
3 v_f^2 = v_i^2 + 2a h Using the kinematic equation to find the final velocity, where v_i is initial velocity, v_f is final velocity, a is acceleration and h is displacement.
4 a = -9.8 \, \text{m/s}^2 Acceleration due to gravity (negative since it’s downward).
5 v_f^2 = (20 \, \text{m/s})^2 + 2(-9.8 \, \text{m/s}^2)(-42 \, \text{m}) Substitute the known values into the kinematic equation.
6 v_f^2 = 400 + 823.2 Calculate the expression inside the equation.
7 v_f = \sqrt{1223.2} \, \text{m/s} Taking the square root of both sides to solve for the final velocity v_f.
8 v_f \approx 35 \, \text{m/s} Final speed of the rock just before it falls into the river.

# (b) Time Taken for the Rock to Strike the River

Step Derivation/Formula Reasoning
1 v_f = v_i + a t Using the first kinematic equation to solve for time t, where v_f is final velocity, v_i is initial velocity, a is acceleration, and t is time.
2 v_f = -35 \, \text{m/s} Since the rock is falling downward, take final velocity v_f as negative.
3 -35 \, \text{m/s} = 20 \, \text{m/s} + (-9.8 \, \text{m/s}^2) t Substitute the known values into the time equation.
4 -35 = 20 – 9.8 t Simplify the equation.
5 -55 = -9.8 t Combine like terms.
6 t = \frac{55}{9.8} Isolate t by dividing both sides of the equation by -9.8.
7 t \approx 5.6\, \text{s} Solving for time gives approximately 5.6 seconds.

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  1. 35.2 m/s
  2. 5.6 seconds

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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)}
\omega (Angular Velocity)\text{radians per second (rad/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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