Motion II Test Review

1.2: Speed and Velocity

• Speed (v) is a key concept to use when quantifying motion, it's the rate of movement (time rate of change of distance from a reference point).

• It is attained using the formula d/t and expressed in a unit of distance over one of time, for example; meters per second (m/s), kilometers per hour (km/h) or miles per hour (mph).
• It is a scalar quantity, it only has a magnitude (#), no direction.
• Speed has 3 aspects that should be highlighted:
1. Speed is relative (depends on a reference point)
2. It is important to differentiate average speed and instantaneous speed.
• Average speed is the total distance an object travels over the time it took to travel that distance.
• Instantaneous speed is an object's speed at a given instant; to calculate it a very short distance over a very short time is used.  Speedometers give instantaneous speed.
• To calculate the speed of a segment of an object's motion we need to set a starting and ending point to then divide the change in distance (d_f - d_i) over the change in time (t_f - t_i).
• The symbol Δ (delta) is a capital greek letter used to represent a "change in" a physical quantity.
• Change in distance = Δd
• Change in time = Δt
• When speed (v) is constant, distance ∝ time (proportionality) in the formula d = vt.
• ∝ means "is proportional to".
• v would be the constant of proportionality.
• When we say d ∝ t we mean that distance will increase proportionally to time in the formula, for example; if the time is doubled, the distance will also be doubled.
• The speed of light in empty space is the universe's absolute speed limit, it is represented with the letter c.
• Nothing in the world travels faster than c.
• Direction is another important aspect of motion; velocity is a physical quantity that incorporated speed and direction (speed in a particular direction, directed motion).
• If an object's speed stays the same, but its direction changes, its velocity changes,
• You can determine velocity with a speedometer and a compass, a GPS gives velocity.
• Velocity is a vector, it has both magnitude (#) and direction.
• Velocity can be negative if it is going in an opposite direction; the negative would be speaking of direction, not magnitude.
• Vectors, like velocity, are represented by proportional arrows; we can add two of these together to obtain a resultant or net velocity.
• In vector addition, we put the two arrows head-to-tail and then draw a resultant.
• If the vectors are going the same way we add them, if they're going different directions we subtract them and the resultant goes toward the largest magnitude's direction.
• If the vectors are in non-opposite directions we must draw them, head-to-tail, and then use the Pythagorean Theorem (c² = a² + b²) to obtain the hypotenuse's (resultant) length.

THIS SECTION'S REVIEW IS VERY BRIEF BECAUSE IT CAME IN THE LAST TEST.  FOR A MORE IN-DEPTH REVIEW YOU CAN USE THE REVIEW FOR THE FIRST MOTION TEST.  SECTION 1.3 IS ALSO IN THAT REVIEW, EVEN THOUGH IT MAY BE MORE THOROUGH ON THIS ONE.

1.3: Acceleration

• Acceleration is a vector quantity of the rate of change of velocity (Δv/Δt).  It indicates how rapidly velocity is changing and is expressed in a unit of distance over a unit of time squared.
  
• An object is in acceleration when it is:
• speeding up (+)
• slowing down or decelerating (-)
• An object is NOT accelerating when it:
• stands still
• moves at a constant velocity
• Acceleration has the same relation with velocity that velocity has with displacement.
• Acceleration indicates how rapidly velocity is changing.
• Velocity indicates how rapidly displacement is changing.
• When an object's acceleration is 1m/s² it means the object is going 1m/s faster every second.
• If the object's speed was 1m/s at 1s, at 2s it will be 2m/s, at 3s it will be 3m/s...
• When acceleration is negative it means the acceleration is going in the opposite direction of the positive velocity, the object is decelerating or slowing down.
• Free falling bodies are only acted on by the force of gravity, ignoring air resistance, they accelerate at 9.8m/s².  This quantity (9.8m/s²) is called acceleration due to gravity and is represented with the letter g.
• g is often used as a measure of acceleration.  An acceleration of 19.6m/s² equals 2g.
• To convert from m/s² to g we must divide the magnitude in m/s² by 9.8.  We can do this by setting up the conversion factor ^1g⁄_9.8m/s².
• Centripetal (center-seeking) acceleration is the acceleration of an object moving in a circular path.
• It is always perpendicular to the object's velocity.
• Its magnitude depends on speed (v) and the radius (r) of a curve; we use the formula ^v2⁄_r to determine centripetal acceleration.
• Acceleration is proportional to the square of the speed (a ∝ v²); this means that when the speed is doubled the acceleration becomes four times as large.
• Acceleration is inversely proportional to the radius (a ∝ ¹⁄_r.); this means that if the radius is doubled, the acceleration becomes half as large,
• A large radius means a path hat is not sharply curved, so velocity changes more slowly and the acceleration is smaller.
• Occurs when you're in the car and take a sharp turn; if you have books on your lap they will probably shift towards the outside of the curve due to the center-seeking centripetal acceleration the car is undergoing.
• Changing the direction of motion of a body is the same kind of thing as changing its speed; they can cause similar effects.  We see both changes in centripetal acceleration.
• A car's cornering acceleration is often evaluated and given; this is measured by the maximum centripetal acceleration it can have when it rounds a curve,
• 0.85g, or 8.33m/s² would be a typical value of lateral, or cornering acceleration of a sports car.

1.4: Simple Types of Motion

• There are several types of motion we can graph; when graphing:
• Time is always written in the x-axis (horizontal)
• The slope of a graph is a measure of its steepness attained through the formula ^rise⁄_run.
• Speed or velocity is the slope of a distance vs. time graph.
• Acceleration is the slope of a velocity vs. time graph.

1. Zero Velocity / Stationary Object: No motion occurs; the object sits still.  The distance of the object, from a reference point, is constant.  Velocity and acceleration are 0.  In a distance vs. time graph, the line would go flat.
2. Constant Velocity / Uniform Motion: The body moves at a uniform motion with a constant velocity (constant speed in a fixed direction).  Acceleration is 0 because the object's speed is not increasing nor decreasing, it's staying the same.  In a distance vs. time graph, it would be a straight line; an upward slope would mean positive velocity whilst an old one would not be very good.

   

   The horse, the runner, and the walker all have a constant speed because their lines are straight in a distance vs. time graph.  The horse is going the fastest because it has the steeper slope, the runner is the second fastest and the walker would be the slowest.

3. Constant Acceleration / Uniform Acceleration: Velocity is changing at a fixed rate, a freefalling object is in constant acceleration (g).  When an object starts from rest and has a constant acceleration we can use the formula v = at and their derived ones.  In constant acceleration, d ∝ t².  When acceleration is constant there is a straight line in a velocity vs. time graph, because the slope of a v vs, t graph is a.  A parabolic (curved) line in a distance vs, time graph is formed because the speed (slope of a d vs. t graph) is increasing.

   

   In both of these graphs, there is constant acceleration.  In the first one, velocity vs. time, the slope is acceleration, so if it is constant, the line will be straight.  In the second one, distance vs. time, the slope is speed, since the speed is increasing the slope is also increasing, creating a parabolic line.

• Even though graphs are a very good way of showing relations between physical quantities, math is better because graphs are limited to two (or sometimes 3) variables.  Math is abstract and almost like a language.

• The slope of a distance vs. time graph is speed/velocity.
• The slope of a velocity vs. time graph is acceleration.
• In this distance vs. time graph, velocity is the slope; this graph has a varying slope.

At point A the car accelerates & velocity is increasing.
At point B velocity is constant.
At point C the object is standing still & undergoing no motion the slope is 0.
At point D the car is going in an opposite direction, making velocity negative.

Formulas

v = √ ar 

v_i² = v_f² - 2ad

v_i = v_f - at

Pythagorean: a² + b² = c²

Area of a trapezoid: ^(a+b)h⁄₂

THERE MAY BE MISTAKES IN THE FORMULAS.  CHECK THEM WITH YOUR NOTES.

THE REVIEW DOES NOT INCLUDE ALL TEST MATERIAL.  YOU SHOULD ALSO READ FROM THE BOOK AND STUDY YOUR NOTES AND THE TEACHER'S PRESENTATION.  YOU CAN USE THE FIRST MOTION TEST REVIEW SINCE SOME MATERIAL IS REVIEWED THERE AS WELL.

References

Ostdiek, Vern J., and Donald J. Bord. Inquiry into Physics. Pacific Grove, CA: Brooks/Cole, 2000. WebAssign. Web. 10 Oct. 2016.