Chapters 1-3 Exam Sheet
What to include when solving problems
Reference frame
Picture
List given variables
List what you need to find
Pure equations (no substitutions)
Step by step solutions
Units
Show unit cancelations
Variables
Variable | Meaning | Vector or Scalar | Units |
---|---|---|---|
m | Mass | Scalar | kg |
x | Displacement on x-axis | Vector | m |
y | Displacement on y-axis | Vector | m |
v | Velocity | Vector | m/s |
t | Time | Scalar | s |
a | Acceleration | Vector | m/s² |
θ | Angle Theta | Scalar | Degrees |
Units Prefixes to Denote Multiples of Ten
Prefix | Symbol | Factor |
---|---|---|
mega | M | 106 |
kilo | k | 103 |
centi | c | 10-2 |
milli | m | 10-3 |
micro | u | 10-6 |
nano | n | 10-9 |
1 megameter = 1000000 m
1 kilometer = 1000 m
1 centimeter = 0.01 m
1 millimeter = 0.001 m
1 micrometer = 0.000001 m
1 nanometer = 0.000000001 m
Converting Units
SI Units
Meters
Kilograms
Seconds
Other conversions
1 mile = 1609.34 meters =5280 ft = 1.60934 km
1 meter = 3.281 ft
1 inch = 2.54 cm
1 hour = 3600 s
1 kilometer = 1000 m
Chapter 1: Introduction and Mathematical Concepts
Trigonometry
Variables for Functions
h = hypothenuse
o = opposite
a = adjacent
sin = y-axis
cos = x-axis
R = Resultant
Functions
sinθ=ho
cosθ=ha
tanθ=ao
Inverse Function
θ=sin−1(ho)
θ=cos−1(ha)
θ=tan−1(ao)
Pythagorean Theorem/Finding Resultant
h2=o2+a2
R2=A2+B2
Vocabulary
Scalars - size or magnitude, described by single number
Temperature
Speed
Mass
Volume
Time
Density
Energy
Theta
Vectors - has magnitude AND direction, represented by arrows
Velocity (speed with direction)
Displacement
Acceleration
Force
Chapter 2: One Dimensional Kinematics
Average speed - how fast something travels over a certain distance
Average speed = Distance/Elapsed time
Change in: Δ = finial - initial
Constant speed - constant velocity (constant magnitude and direction)
Average velocity - how fast an object's position changes over time
v=ΔtΔx
Instantaneous velocity - velocity of an object at a specific moment in time, measures both the speed and direction of motion of the object at that precise instant.
v=limΔt→0ΔtΔx
Average acceleration rate at which an object's velocity changes over a certain period of time.
a=ΔtΔv
Displacement: Δx=x−x0
Slope: = rise/run = x2−x1y2−y1
Speed vs. Velocity vs. Acceleration
Speed: How fast you're going, like on a speedometer. It's just a number, no direction.
Velocity: Speed with direction. It's how fast you're going and in which direction.
Acceleration: Changes in velocity. Speeding up, slowing down, or changing direction.
Velocity and Acceleration
Velocity, no acceleration = constant velocity
Velocity with acceleration = speeds up as it moves
When acceleration and velocity have opposite directions, the object slows down and decelerates.
Instantaneous velocity and instantaneous acceleration:
If inst. velocity goes from positive to negative, then acceleration must be negative.
Kinematic Variables
x = displacement
a = acceleration
v = velocity
vo = initial velocity
t = time
Kinematic Equations
v=vo+at
x=21(vo+v)t
x=vot+21at2
v2=vo2+2ax
When solving these problems:
Assume acceleration is constant
Start motion at origin
Start clock at time = 0
Freely Falling Bodies
In absence in air resistance, all bodies fall vertically with the same acceleration.
Acceleration always points down.
Acceleration of freely falling body is call the acceleration due to gravity.
a = -g
a = -9.80 m/s²
g = 9.80 m/s²
Earth’s free-fall gravitational acceleration.
Scalar quantity
At its maximum height, velocity = 0m/s
Steps:
Draw picture.
Identify known/unknown variables.
Find needed kinematic equations.
Solve.
Free Fall Steps:
Identify known quantities: initial velocity (if any), the acceleration due to gravity (usually -9.8 m/s2), and the time of motion (if given).
Determine the unknown quantities
Apply appropriate equation: x=vot+21gt
Chapter 3
r and x are the same
Displacement: Δr=r−ro
Average velocity: v=ΔtΔr
Instantaneous velocity: v=limΔt→0ΔtΔr
Instantaneous acceleration: a=limΔt→0ΔtΔv
Projectile Motion
Under the influence of gravity alone, an object near the surface of the Earth will accelerate downwards at 9.80 m/s²
ay=−9.80m/s2
ax=0m/s2
vx=vox = constant (same for both spaces)
At the peak, vx=0m/s
Initial velocity:
y-axis: sinθ=vovoy
x-axis: cosθ=vovox
Magnitude of velocity: v2=vx2+vy2
Angle for velocity vectors: tanθ=vxvy
Range:
R=voxt
OR
R=(vocosθ)t
When creating x-component and y-component table for kinematic variables, remember to include directions = (+) or (-).
Vector magnitude is always positive or zero.
Maximum distance an object will go is at a 45 degree angle
Steps:
Identify known/unknown variables.
Separate x and y components.
Find needed kinematics equations.
Find maximum height.
Determine range.
Relative Velocity
Relative velocity refers to the velocity of an object or observer in relation to another object or observer. It is the difference in velocities between two objects or observers as they move in different directions or at different speeds. Relative velocity takes into account the motion of both objects or observers and is often used to analyze the motion of objects in different frames of reference.
The velocity of object A relative to object B is vAB , and the velocity of object B relative to object C is vBC
The velocity of A relative to C is shown in the equation
vAC=vAB+vBC
The velocity of object R relative to object S is always the negative of the velocity of object S relative to R vRS=−vSR
Kai’s AI response:
To solve a relative velocity problem, follow these steps:
Identify the objects involved: Determine the two objects or individuals whose velocities are being compared.
Define the frame of reference: Choose a reference point or frame of reference from which the velocities will be measured.
Determine the velocities: Find the velocities of the objects or individuals with respect to the chosen frame of reference. This may involve considering their initial velocities, accelerations, and any other relevant factors.
Determine the relative velocity: Calculate the relative velocity by subtracting the velocity of one object from the velocity of the other. Pay attention to the direction of motion.
Solve for the desired quantity: Depending on the problem, you may be asked to find the relative velocity, time of encounter, distance traveled, or any other related quantities. Use the given information and equations of motion to solve for the desired quantity.
Check your answer: Once you have obtained a solution, double-check your calculations and ensure that your answer makes sense in the context of the problem.
Remember to use appropriate units and consider any assumptions or simplifications made during the problem-solving process.
Extra
Average speed - how fast something travels over a certain distance: distance/elapsed time
Average velocity - how fast an object’s position changes over time: v=ΔtΔx
Average acceleration - rate at which an object’s velocity changes over a period of time: a=ΔtΔv
Speed vs. Velocity vs. Acceleration
Speed: How fast you're going, like on a speedometer. It's just a number, no direction.
Velocity: Speed with direction. It's how fast you're going and in which direction.
Acceleration: Changes in velocity. Speeding up, slowing down, or changing direction.
Velocity and Acceleration
Velocity, no acceleration = constant velocity
Velocity with acceleration = speeds up as it moves
When acceleration and velocity have opposite directions, the object slows down and decelerates.
Chapters 1-3 Exam Sheet
What to include when solving problems
Reference frame
Picture
List given variables
List what you need to find
Pure equations (no substitutions)
Step by step solutions
Units
Show unit cancelations
Variables
Variable | Meaning | Vector or Scalar | Units |
---|---|---|---|
m | Mass | Scalar | kg |
x | Displacement on x-axis | Vector | m |
y | Displacement on y-axis | Vector | m |
v | Velocity | Vector | m/s |
t | Time | Scalar | s |
a | Acceleration | Vector | m/s² |
θ | Angle Theta | Scalar | Degrees |
Units Prefixes to Denote Multiples of Ten
Prefix | Symbol | Factor |
---|---|---|
mega | M | 106 |
kilo | k | 103 |
centi | c | 10-2 |
milli | m | 10-3 |
micro | u | 10-6 |
nano | n | 10-9 |
1 megameter = 1000000 m
1 kilometer = 1000 m
1 centimeter = 0.01 m
1 millimeter = 0.001 m
1 micrometer = 0.000001 m
1 nanometer = 0.000000001 m
Converting Units
SI Units
Meters
Kilograms
Seconds
Other conversions
1 mile = 1609.34 meters =5280 ft = 1.60934 km
1 meter = 3.281 ft
1 inch = 2.54 cm
1 hour = 3600 s
1 kilometer = 1000 m
Chapter 1: Introduction and Mathematical Concepts
Trigonometry
Variables for Functions
h = hypothenuse
o = opposite
a = adjacent
sin = y-axis
cos = x-axis
R = Resultant
Functions
sinθ=ho
cosθ=ha
tanθ=ao
Inverse Function
θ=sin−1(ho)
θ=cos−1(ha)
θ=tan−1(ao)
Pythagorean Theorem/Finding Resultant
h2=o2+a2
R2=A2+B2
Vocabulary
Scalars - size or magnitude, described by single number
Temperature
Speed
Mass
Volume
Time
Density
Energy
Theta
Vectors - has magnitude AND direction, represented by arrows
Velocity (speed with direction)
Displacement
Acceleration
Force
Chapter 2: One Dimensional Kinematics
Average speed - how fast something travels over a certain distance
Average speed = Distance/Elapsed time
Change in: Δ = finial - initial
Constant speed - constant velocity (constant magnitude and direction)
Average velocity - how fast an object's position changes over time
v=ΔtΔx
Instantaneous velocity - velocity of an object at a specific moment in time, measures both the speed and direction of motion of the object at that precise instant.
v=limΔt→0ΔtΔx
Average acceleration rate at which an object's velocity changes over a certain period of time.
a=ΔtΔv
Displacement: Δx=x−x0
Slope: = rise/run = x2−x1y2−y1
Speed vs. Velocity vs. Acceleration
Speed: How fast you're going, like on a speedometer. It's just a number, no direction.
Velocity: Speed with direction. It's how fast you're going and in which direction.
Acceleration: Changes in velocity. Speeding up, slowing down, or changing direction.
Velocity and Acceleration
Velocity, no acceleration = constant velocity
Velocity with acceleration = speeds up as it moves
When acceleration and velocity have opposite directions, the object slows down and decelerates.
Instantaneous velocity and instantaneous acceleration:
If inst. velocity goes from positive to negative, then acceleration must be negative.
Kinematic Variables
x = displacement
a = acceleration
v = velocity
vo = initial velocity
t = time
Kinematic Equations
v=vo+at
x=21(vo+v)t
x=vot+21at2
v2=vo2+2ax
When solving these problems:
Assume acceleration is constant
Start motion at origin
Start clock at time = 0
Freely Falling Bodies
In absence in air resistance, all bodies fall vertically with the same acceleration.
Acceleration always points down.
Acceleration of freely falling body is call the acceleration due to gravity.
a = -g
a = -9.80 m/s²
g = 9.80 m/s²
Earth’s free-fall gravitational acceleration.
Scalar quantity
At its maximum height, velocity = 0m/s
Steps:
Draw picture.
Identify known/unknown variables.
Find needed kinematic equations.
Solve.
Free Fall Steps:
Identify known quantities: initial velocity (if any), the acceleration due to gravity (usually -9.8 m/s2), and the time of motion (if given).
Determine the unknown quantities
Apply appropriate equation: x=vot+21gt
Chapter 3
r and x are the same
Displacement: Δr=r−ro
Average velocity: v=ΔtΔr
Instantaneous velocity: v=limΔt→0ΔtΔr
Instantaneous acceleration: a=limΔt→0ΔtΔv
Projectile Motion
Under the influence of gravity alone, an object near the surface of the Earth will accelerate downwards at 9.80 m/s²
ay=−9.80m/s2
ax=0m/s2
vx=vox = constant (same for both spaces)
At the peak, vx=0m/s
Initial velocity:
y-axis: sinθ=vovoy
x-axis: cosθ=vovox
Magnitude of velocity: v2=vx2+vy2
Angle for velocity vectors: tanθ=vxvy
Range:
R=voxt
OR
R=(vocosθ)t
When creating x-component and y-component table for kinematic variables, remember to include directions = (+) or (-).
Vector magnitude is always positive or zero.
Maximum distance an object will go is at a 45 degree angle
Steps:
Identify known/unknown variables.
Separate x and y components.
Find needed kinematics equations.
Find maximum height.
Determine range.
Relative Velocity
Relative velocity refers to the velocity of an object or observer in relation to another object or observer. It is the difference in velocities between two objects or observers as they move in different directions or at different speeds. Relative velocity takes into account the motion of both objects or observers and is often used to analyze the motion of objects in different frames of reference.
The velocity of object A relative to object B is vAB , and the velocity of object B relative to object C is vBC
The velocity of A relative to C is shown in the equation
vAC=vAB+vBC
The velocity of object R relative to object S is always the negative of the velocity of object S relative to R vRS=−vSR
Kai’s AI response:
To solve a relative velocity problem, follow these steps:
Identify the objects involved: Determine the two objects or individuals whose velocities are being compared.
Define the frame of reference: Choose a reference point or frame of reference from which the velocities will be measured.
Determine the velocities: Find the velocities of the objects or individuals with respect to the chosen frame of reference. This may involve considering their initial velocities, accelerations, and any other relevant factors.
Determine the relative velocity: Calculate the relative velocity by subtracting the velocity of one object from the velocity of the other. Pay attention to the direction of motion.
Solve for the desired quantity: Depending on the problem, you may be asked to find the relative velocity, time of encounter, distance traveled, or any other related quantities. Use the given information and equations of motion to solve for the desired quantity.
Check your answer: Once you have obtained a solution, double-check your calculations and ensure that your answer makes sense in the context of the problem.
Remember to use appropriate units and consider any assumptions or simplifications made during the problem-solving process.
Extra
Average speed - how fast something travels over a certain distance: distance/elapsed time
Average velocity - how fast an object’s position changes over time: v=ΔtΔx
Average acceleration - rate at which an object’s velocity changes over a period of time: a=ΔtΔv
Speed vs. Velocity vs. Acceleration
Speed: How fast you're going, like on a speedometer. It's just a number, no direction.
Velocity: Speed with direction. It's how fast you're going and in which direction.
Acceleration: Changes in velocity. Speeding up, slowing down, or changing direction.
Velocity and Acceleration
Velocity, no acceleration = constant velocity
Velocity with acceleration = speeds up as it moves
When acceleration and velocity have opposite directions, the object slows down and decelerates.