What is kinematics?
-a description of motion
What are the two different types of kinematics?
-scalar and vector
What is a position vector?
-a value that has a numerical value and a direction, where the tail of the vector is at the origin and the head is at the location of the object
What mathematical symbol denotes position vector?
x with a hat
Can vectors have a (+) or (-) sign associate with them?
-yes
What is the tail of a position vector always tied to?
-the origin
Where is the head of a position vector located?
-at the final location of an object
Where is the tail of a displacement vector located?
-at the initial location of an object
Where is the head of a displacement vector located?
-at the final location of an object
What mathematical symbol denotes displacement vector?
?x with a hat
If the tail of a displacement vector is at the initial location of an object and the head is at the final location of an object, how is ?x calculated?
?x = final position - initial position
?x = xf - xi
Is displacement origin dependent? Why or why not?
-no, displacement is not origin dependent
-a result of the fact that displacement is calculated using two points, neither of which are at the origin
Do displacements have a (+) or a (-) sign associated with them?
-yes, they do
How does distance differ from displacement?
-displacement accounts for the "net effect" of the trip, and throws away all details
-however, distance accounts for "every stop along the way
As such, how do distance and displacement typically compare numerically?
d? |?x| (delta x with a hat)
Can distance have a (+) or a (-) sign associated with them?
-no they cannot
As such, what is the only thing that distances have associated with them?
-a numerical value
Are average speed and average velocity the same thing?
-no they are not
Why not?
-because speed is scalar and velocity is a vector
If average speed is scalar, does it have a (+) or a (-) sign associated with it?
-no it does not, it is always positive
Why specifically is average speed always positive?
-because both distance and elapsed time are positive
What is average speed equal to?
Average speed = distance / ? time
What is average velocity equal to?
Average velocity = ?x/?t = (x final-x initial)/(t final-t initial)
The sign of what will be matched by the sign of the average velocity?
-displacement
What does the average velocity tell us?
-the direction that an object is moving in
What values can average velocity have?
-average velocity can be positive, negative, or zero
If average velocity is greater than zero, what direction does the object move in?
-the positive direction
If average velocity is less than zero, what direction does the object move in?
-the negative direction
If average velocity equals zero, what direction does the object move in?
-the object is stationary, so it does not move in the positive or negative direction
Why is average velocity used more frequently than average speed?
-because it provides us the direction that the object is moving in
How many dimensions are associated with displacement?
-only one
How many dimensions are associated with velocity?
-only one
How are average velocity and average speed related to each other?
average speed ? |average velocity|
What does this mean?
-average speed will be greater than or equal to average velocity
Why?
-because d? |?x| (delta x with a hat)
What is instantaneous velocity?
-the velocity of an object at an isolated, small increment of time
What is instantaneous speed?
-the speed of an object at an isolated, small increment of time
What is the relationship between instantaneous speed and velocity?
instantaneous speed = |instantaneous velocity|
What is the only difference between instantaneous speed and instantaneous velocity?
-speed will always be positive, while velocity can be positive, negative, or zero
What are the two methods that can be used to calculate average speed?
-use distance / ? time
-average all instantaneous speeds to find average speed, using time averages
What are the two methods that can be used to calculate average velocity?
-use ?x/?t
-average all instantaneous velocities to find average velocity, using time averages
What is the definition of acceleration?
-the rate of change of velocity over time
-how many meters per second the velocity changes by every second
-the set of instructions that tells the velocity how to change with time
-how acceleration evolves over time
What is the mathematical expression of acceleration?
?v/?t = (v final-v initial)/(t final-t initial)
Is acceleration a vector?
-yes
-it has both a magnitude and a direction
What values can acceleration have?
-can be positive, negative, or zero
How many dimensions are associated with acceleration?
-one dimension
Do acceleration and velocity have the same sign? Opposite signs?
-acceleration and velocity may have the same or opposite signs
What is the effect on an object when acceleration and velocity have the same sign?
-the speed of the object increases
-the velocity of the object increases in the (+) direction
Does velocity increase uniformly with a positive acceleration?
-not necessarily
When does velocity increase uniformly with a positive acceleration?
-when acceleration is constant
When does velocity not increase uniformly with a positive acceleration?
-when acceleration is not constant
What is acceleration equal to when velocity is constant?
-ZERO
What is the effect on an object when velocity is positive and acceleration is negative?
-the object will slow down as acceleration "chips away" at its speed and positive velocity
-eventually, the object will come to a complete standstill, then will begin to travel with greater velocity in the negative direction
-->first, the acceleration chi
Suppose that we are measuring the movement of the object in seconds. If negative acceleration is applied to a positive velocity, will the distance covered during the first second of negative acceleration be greater than or less than the distance covered d
-the distance covered during the first second will be greater
What is the effect on an object when velocity is negative and acceleration is negative?
-the speed of the object increases
-the velocity of the object increases in the negative direction
What is the effect on an object when velocity is negative and acceleration is positive?
-the object will slow down as acceleration "chips away" at its speed and negative velocity
-eventually the object will come to a complete standstill, then will begin to travel with greater velocity in the positive direction
-->first, the acceleration chip
Can you be at rest and still have an acceleration?
-YES!
-acceleration can be negative or positive
-you won't be staying at rest for very long
When are instantaneous speed and velocity the same?
-when velocity is constant
When are instantaneous acceleration and average acceleration the same?
-when acceleration is constant
What are the constant acceleration equations?
-a set of equations that govern the kinematics of constant acceleration
When should these equations never be applied?
-when acceleration is not constant
Does position have to be continuous? Why or why not?
-yes, it has to be continuous
-if you want to move from one position to the other, you must pass through all intermediate positions
Does velocity have to be continuous? Why or why not?
-yes it has to be continuous
-if you want to move from one velocity to the other, you must pass through all intermediate velocities
Does acceleration have to be continuous? Why or why not?
-no, it does not have to be continuous
-one can change the instructions to velocity instantaneously
What is gravity?
-the constant force of acceleration on objects in free fall
(Provided what can be ignored)?
-air resistance
When an object is travelling "up," what will gravity do?
-chips away 9.81 m/s from its velocity every second
When an object is travelling "down," what will gravity do?
-adds 9.81 m/s to its velocity every second
Is gravity necessarily negative? Why or why not?
-no, it is not necessarily negative
-it is dependent upon the mathematical scaffolding that is setup
When can gravity be used?
-only when an object is near the surface of Earth
What does "near the surface of Earth" mean exactly?
-equal to anywhere you will go in your entire lifetime (except if you travelled into space)
-everyday conditions
When can g not be used?
-if an object is held in place by hand or other tool
-if an object has hit the ground (or otherwise been stopped)
-if air resistance is not negligible
Can the final velocity in the y direction of an object be equal to zero when an object hits the ground?
-no it cannot, because gravity is imposing a constant acceleration on an object
-for this reason, final velocity is assumed to be the moment right before an object hits the ground
When acceleration is constant, does it ever change direction? Why or why not?
-no it does not, because it is constant
-it is only velocity that changes direction
When an object is released from a hand or a vessel (such as a hot air balloon), what is its velocity?
-the velocity of the object is the same as the velocity of the hand or vessel from which it was reelased
When an object is in a flight with positive y displacement, what is its velocity when it as at the top of its motion? Why?
-velocity is equal to zero
-it is changing direction, moving from a positive upwards flight to a negative downwards flight
What is a ground level flight?
-when an object enters and exits flight at the same height
What two important things can be assumed for a ground level flight?
-time up is equal to time down
-velocity final = (-)velocity initial
Why is time up equal to time down?
-because the same acceleration is governing the motion of the object
What mathematical problem/hitch does equation number three pose?
-a positive and a negative value can be obtained for final velocity
How then, do you determine if final velocity is positive or negative?
-use intuition
-if an object is travelling downward, then final velocity cannot be positive
When an object's velocity is zero, does its acceleration have to be zero?
-NO!
-an objects velocity cannot stay at zero, it must pass through it
What does the area underneath a position vs. time graph represent?
-the area under the graph does not represent anything physical
What does the slope of a position vs. time graph represent?
-velocity
How is velocity calculated in this case?
-using rise/run
-rise/run = ?x/?t = (x final-x initial)/(t final-t initial)
What does a positive slope on a position versus time graph indicate?
-that velocity is positive
-object is travelling in a (+) x direction over time
What does a negative slope on a position versus time graph indicate?
-that velocity is negative
-object is travelling in a (-) x direction over time
What does a zero slope on a position versus time graph indicate?
-that velocity is equal to zero
-object is stationary
What does the area under a velocity vs. time graph represent?
-displacement, ?x
What does the slope of a velocity vs. time graph represent?
-average acceleration
How is acceleration calculated in this case?
-using rise/run
-rise/run = ?v/?t = (v final-v initial)/(t final-t initial)
What does a positive slope on a velocity versus time graph indicate?
-that acceleration is positive
-object is increasing the velocity at which it travels over time
What does a negative slope on a velocity versus time graph indicate?
-that acceleration is negative
-object is decreasing the velocity at which it travels over time
What does a zero slope on a velocity versus time graph indicate?
-that acceleration is zero
-the velocity at which an object is traveling remains constant
Can acceleration be negative and movement in the x direction be positive?
-yes, if velocity is still positive
On a velocity versus time graph, what do we know if velocity is greater than 0?
-that movement of the object is still in the positive x direction
On a velocity versus time graph, what do we know if velocity is equal to zero?
-that the object is at rest
On a velocity versus time graph, what do we know if velocity is less than zero?
-that movement of the object is in the negative x direction
What does the area under an acceleration vs. time graph represent?
-change in velocity, ?v
On an acceleration versus time graph, what do we know if acceleration is greater than 0? In terms of a, v, and x?
-acceleration is positive
-velocity may be positive or negative
-object may be moving in a positive or negative x direction
On an acceleration versus time graph, what do we know if acceleration is equal to 0? In terms of a, v, and x?
-acceleration is equal to zero
-velocity is constant
-object may be moving in a positive or negative x direction
On an acceleration versus time graph, what do we know if acceleration is less than 0? In terms of a, v, and x?
-acceleration is negative
-velocity may be positive or negative
-object may be moving in a positive or negative x direction
Of the position v time, the velocity v. time, and the acceleration v. time, which can be curved?
-only the position v. time graph
On a distance vs. time graph, when must the graph be curved?
-if something is changing velocity: if object is speeding up or slowing down
-if an object is speeding up, in a given period of time you have to cover more distance at the end of the interval than at the beginning of the interval
Is it possible to drive your car in such a way that the distance it covers is less than the magnitude of its displacement? Why or why not?
-no, it is not possible to do so
-any deviation from a straight line results in a distance that is greater than the magnitude of the displacement
Assume that the brakes in your car create a constant deceleration, regardless of how fast you are going. If you double your driving speed, how does this affect the time required to come to a stop?
-time required to stop doubles
-(can plug in values to the constant acceleration equations)
If you double your driving speed, how does this affect the distance required to come to a stop?
-distance required to stop quadruples
-(can plug in values to the constant acceleration equations)
After winning a baseball game, one player drops a glove while another tosses a glove into the air. how do the acceleration of the two gloves compare?
-accelerations are equal
How do you deduce velocity from a position versus time graph?
-velocity is equal to the slope of the position versus time curve
You drive in a straight line at 15 m/s for 10 minutes, then at 25 m/s for 10 minutes. What is your average speed? Why?
-average speed equals 20 m/s
-you spend equal amounts of time at 15 m/s and 25 m/s
In what two types of scenarios can we expect an object to come to rest at some point in time during its motion? Why?
-when a < 0 and v > 0
-when a > 0 and v < 0
-->because the acceleration and the velocity vector are in opposite directions
You drive 4.0 mi at 30.0 mi/h and then another 4.0 mi at 50 mi/h. Is your average speed for the 8 mile trip greater, equal to, or less than 40.0 mi/h? Why?
-less than 40 mi/hr because it takes a longer period of time to drive 30 mi/h than 50 mi/h
You drop a rock from a bridge to the river below. When the rock has fallen 4 m, you drop a second rock. As the rocks continue their fall, does their separation increase, decrease, or stay the same? Why?
-the separation between the rocks increases
-the rock that has a head start always has a greater velocity than the later one; thus it covers a greater distance in any interval of time. thus, the separation between the rocks increases.
At the edge of a roof you drop ball A from rest, then throw ball B downward with an initial velocity. Is the increase in speed just before the balls land more for ball A, more for ball B, or the same for each ball?
-more for ball A, because ball A has more time to reach the ground than ball B
A ball is thrown straight up with an initial speed. When it reaches the top of its flight at height h, a second ball is thrown with the same initial speed. Do the balls cross paths at height 1/2h, above 1/2h, or below 1/2h?
-the initial speed of the first ball (at height h) is zero; the initial speed of the second ball is greater than zero
-therefore, the second ball covers more distance in a given amount of time, and the balls meet at a height greater than h/2
An automobile driver applies the brakes and slows to a stop with constant acceleration. After half the stopping distance, is the automobile's speed less than, equal to, or greater than 0.5velocity initial? Why?
-greater than 0.5 velocity initial
-stopping distance for the automobile depends on velocity initial^2. thus, ?x increases by a factor of 4 when speed is doubled, and is equal to ?x/4 when speed is reduced by 1/2j
The average velocity of an object for a given motion points to the right. Is it possible for the instantaneous velocity to point to the left at some point in the middle of the motion?
-yes
At some point at the beginning of the motion?
-yes
At some point at the end of the motion?
-yes
When are x vs. t graphs curved?
-under constant (non-zero) acceleration
What are the implications of this in terms of velocity?
-velocity does not increase and decrease uniformly with distance
True or false: a trip with an average velocity to the right must have an instantaneous velocity to the right during the entire trip. Why?
-FALSE
-average velocity just means that the object must travel more right than left overall
-at any given moment during the trip, the object could be going left
True or false: If an object has constant acceleration, it also has a constant velocity. Why?
-FALSE
-a constant (non-zero) acceleration means a smoothly changing velocity, not a constant velocity
What is the only case in which a constant acceleration produces a constant velocity?
-a constant acceleration of zero
True or false: If an object has a constant speed in a straight line, it has a constant velocity? Why?
-TRUE
-Constant speed, constant direction (straight line) means the velocity (which includes both speed and direction) must be constant
A car travels in a straight line, covering a total distance of 90 miles in 60 minutes. Is the velocity of the car constant?
-no
-the car could be speeding up or slowing down and still travel 90 miles in 60 minutes
Is the acceleration of the car non-zero?
-no
-the car could be travelling at constant velocity and still travel 90 miles in 60 minutes
Were the first 45 miles covered in 30 minutes?
-no
-the car would only cover half the distance in half the time if it was travelling at a constant velocity
Was the average velocity of the car 90 miles per hour in the direction of motion?
-yes
Can the acceleration of an object be zero while the velocity is not?
-yes, this corresponds to constant velocity (constant speed, straight line) motion
Can the velocity of an object be constant while the acceleration is not?
-no, this is not possible
-acceleration measures a change in velocity, so if the velocity is constant the acceleration is zero
Can the acceleration of an object be constant, but the velocity is not?
-yes, this is possible
-if an object has a constant acceleration, this means that the velocity is changing, just doing so in a smooth way