One-Dimensional Kinematics Review
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Part A: Multiple TRUE/FALSE:
- A vector is a large quantity and a scalar is a small quantity.
- A scalar quantity has a magnitude and a vector quantity does not.
- A vector quantity is described with a direction and a scalar is not.
- Scalar quantities are path dependent quantities and vector quantities are not.
- A scalar quantity depends only upon the initial and final values of the quantity; this is not the case for vector quantities.
- The quantity 20 m/s, north is a speed and as such is a scalar quantity.
- The quantity 9.8 m/s/s is an acceleration value and as such is a vector quantity.
a. FALSE - This would never be the case. Vectors simply are direction-conscious, path-independent quantities which depend solely upon the initial and final state of an object. Vectors are always expressed fully by use of a magnitude and a direction.
b. FALSE - Both scalar and vector quantities have a magnitude or value expressed with a given unit; additionally, a vector quantity requires a direction in order to fully express the quantity.
c. TRUE - Vectors are fully described by magnitude AND direction; scalars are not described with a direction.
d. TRUE - Scalars such as distance would depend upon the path taken from initial to final location. If you run around the track one complete time, your distance will be different than if you take a step forward and a step backwards. The path MATTERS; distance (like all scalars) depends upon it. On the other hand, the displacement (a vector quantity) is the same for both paths.
e. FALSE - Vectors are the types of quantities which depend only upon initial and final state of the object. For instance, the vector quantity displacement depends only upon the starting and final location.
f. FALSE - This is certainly not a speed quantity; though the unit is appropriate for speed, the statement of the direction is inconsistent with speed as a scalar quantity.
g. FALSE (a rather picky FALSE) - If a direction was included, then this would be an acceleration value. The unit is characteristic of acceleration but the lack of direction is inconsistent with acceleration being a vector quantity.
Vectors and Scalars
- Distance is a vector quantity and displacement is a scalar quantity.
- A person makes a round-trip journey, finishing where she started. The displacement for the trip is 0 and the distance is some nonzero value.
- A person starts at position A and finishes at position B. The distance for the trip is the length of the segment measured from A to B.
- If a person walks in a straight line and never changes direction, then the distance and the displacement will have exactly the same magnitude.
- The phrase "20 mi, northwest" likely describes the distance for a motion.
- The phrase "20 m, west" likely describes the displacement for a motion.
- The diagram below depicts the path of a person walking to and fro from position A to B to C to D. The distance for this motion is 100 yds.
- For the same diagram below, the displacement is 50 yds.
a. FALSE - Distance is the scalar and displacement is the vector. Know this one!
b. TRUE - Displacement is the change in position of an object. An object which finishes where it started is not displaced; it is at the same place as it started and as such has a zero displacement. On the other hand, the distance is the amount of ground which is covered. And if it was truly a journey, then there is definitely a distance.
c. FALSE - This would only be the case if the person walk along a beeline path from A to B. But if the person makes a turn and veers left, then right and then ..., then the person has a distance which is greater than the length of the path from A to B. Distance refers to the amount of ground which is covered.
d. TRUE - If a person never changes direction and maintains the same heading away from the initial position, then every step contributes to a change in position in the same original direction. A 1 m step will increase the displacement (read as out of place-ness) by 1 meter and contribute one more meter to the total distance which is walked.
e. FALSE - Distance is a scalar and is ignorant of direction. The "northwest" on this quantity would lead one to believe that this is a displacement (a vector quantity) rather than a distance.
f. TRUE - The unit is an appropriate displacement unit (length units) and the direction is stated. Since there is both magnitude and direction expressed, one would believe that this is likely a displacement.
g. FALSE - The distance from A to B is 35 yds; from B to C is 20 yds; and from C to D is 35 yds. The total distance moved is 90 yds.
h. FALSE (a rather picky FALSE) - Technically, this is not a displacement since displacement is a vector and fully described by both magnitude and direction. The real expression of displacement is 50 yds, left (or west or -)
Distance vs. Displacement
- Velocity is a vector quantity and speed is a scalar quantity.
- Both speed and velocity refer to how fast an object is moving.
- Person X moves from location A to location B in 5 seconds. Person Y moves between the same two locations in 10 seconds. Person Y is moving with twice the speed as person X.
- The velocity of an object refers to the rate at which the object's position changes.
- For any given motion, it is possible that an object could move very fast yet have an abnormally small velocity.
- The phrase "30 mi/hr, west" likely refers to a scalar quantity.
- The average velocity of an object on a round-trip journey would be 0.
- The direction of the velocity vector is dependent upon two factors: the direction the object is moving and whether the object is speeding up or slowing down.
- The diagram below depicts the path of a person walking to and fro from position A to B to C to D. The entire motion takes 8 minutes. The average speed for this motion is approximately 11.3 yds/min.
- For the same diagram below, the average velocity for this motion is 0 yds/min.
a. TRUE - Yes! Speed is a scalar and velocity is the vector. Know this one!
b. FALSE - Speed refers to how fast an object is moving; but velocity refers to the rate at which one's motion puts an object away from its original position. A person can move very fast (and thus have a large speed); but if every other step leads in opposite directions, then that person would not have a large velocity.
c. FALSE - Person Y has one-half the speed of Person X. If person Y requires twice the time to do the same distance, then person Y is moving half as fast.
d. TRUE - Yes! That is exactly the definition of velocity - the rate at which velocity changes.
e. TRUE - An Indy Race car driver is a good example of this. Such a driver is obviously moving very fast but by the end of the race the average velocity is essentially 0 m/s.
f. FALSE - The presence of the direction "west" in this expression rules it out as a speed expression. Speed is a scalar quantity and direction is not a part of it.
g. TRUE - For a round trip journey, there is no ultimate change in position. As such, the average velocity is 0 m/t seconds. Regardless of the time, the average velocity will be 0 m/s.
h. FALSE - The direction of the velocity vector depends only upon the direction that the object is moving. A westward moving object has a westward velocity.
i. TRUE - As discussed in #2g, the distance traveled is 90 meters. When divided by time (8 minutes), the average speed is 11.25 yds/min.
j. FALSE - The average velocity would be 0 yds/min only if the person returns to the initial starting position. In this case, the average velocity is 50 yds/8 min, west (6.25 yds/min, west).
Speed vs. Velocity
- Acceleration is a vector quantity.
- Accelerating objects MUST be changing their speed.
- Accelerating objects MUST be changing their velocity.
- Acceleration units include the following; m/s2, mi/hr/sec, cm/s2, km/hr/m.
- The direction of the acceleration vector is dependent upon two factors: the direction the object is moving and whether the object is speeding up or slowing down.
- An object which is slowing down has an acceleration.
- An object which is moving at constant speed in a circle has an acceleration.
- Acceleration is the rate at which the velocity changes.
- An object that is accelerating is moving fast.
- An object that is accelerating will eventually (if given enough time) be moving fast.
- An object that is moving rightward has a rightward acceleration.
- An object that is moving rightward and speeding up has a rightward acceleration.
- An object that is moving upwards and slowing down has an upwards acceleration.
Answer: ACEFGHL (and maybe J)
a. TRUE - Yes it is. Acceleration is direction-conscious.
b. FALSE - Accelerating objects could be changing their speed; but it is also possible that an accelerating object is only changing its direction while maintaining a constant speed. The race car drivers at Indy might fit into this category (at least for certain periods of the race).
c. TRUE - Accelerating object MUST be changing their velocity -either the magnitude or the direction of the velocity.
d. FALSE - The first three sets of units are acceleration units - they include a velocity unit divided by a time unit. The last set of units is a velocity unit divided by a length unit. This is definitely NOT an acceleration.
e. TRUE - This is the case and something important to remember. Consider its application in the last three parts of this question.
f. TRUE - Accelerating objects are either slowing down, speeding up or changing directions.
g. TRUE - To move in a circle is to change one's direction. As such, there is a change in the velocity (not magnitude, but the direction part); this constitutes an acceleration.
h. TRUE - This is the very definition of acceleration. Know this one - its the beginning point of all our thoughts about acceleration.
i. FALSE - Accelerating objects are not necessarily moving fast; they are merely changing how fast they are moving (or the direction they are moving).
j. FALSE - If the accelerating object is slowing down, then it will eventually stop and not reach a fast speed. And if that doesn't convince you, then consider an object that is accelerating by moving in a circle at constant speed forever; it will accelerate the entire time but never being going any faster than at the beginning.
k. FALSE - If an object is moving rightward and slowing down, then it would have a leftward acceleration.
l. TRUE - If an object is speeding up, then the direction of the acceleration vector is in the direction which the object is moving.
m. FALSE - If an object is slowing down, then the acceleration vector is directed opposite the direction of the motion; in this case the acceleration is directed downwards.
- Position-time graphs cannot be used to represent the motion of objects with accelerated motion.
- The slope on a position-time graph is representative of the acceleration of the object.
- A straight, diagonal line on a position-time graph is representative of an object with a constant velocity.
- If an object is at rest, then the position-time graph will be a horizontal line located on the time-axis.
- Accelerated objects are represented on position-time graphs by curved lines.
- An object with a positive velocity will be represented on a position-time graph by a line with a positive slope.
- An object with a negative velocity will be represented on a position-time graph by a line with a negative slope.
- An object with a positive acceleration will be represented on a position-time graph by a line which curves upwards.
- An object with a negative acceleration will be represented on a position-time graph by a line which curves downwards.
a. FALSE - Position-time graphs represent accelerated motion by curved lines.
b. FALSE - The slope of a position-time graph is the velocity of the object. Some things in this unit are critical things to remember and internalize; this is one of them.
c. TRUE - A straight diagonal line is a line of constant slope. And if the slope is constant, then so is the velocity.
d. FALSE - Not necessarily true. If the object is at rest, then the line on a p-t graph will indeed be horizontal. However, it will not necessarily be located upon the time axis.
e. FALSE - Accelerating objects (if the acceleration is attributable to a speed change) are represented by lines with changing slope - i.e., curved lines.
f. TRUE - Since slope on a p-t graph represents the velocity, a positive slope will represent a positive velocity.
g. TRUE - Since slope on a p-t graph represents the velocity, a negative slope will represent a negative velocity.
h. FALSE - (This is confusing wording here since we might not all agree on what "curving up" means.) A line that slopes upward and has a curve (perhaps you call that "curving up" as I do) has a positive velocity (due to its positive slope). If the curve is "concave down" (you might say leveling off to a horizontal as time progresses) then the object is slowing down and the acceleration is negative.
i. FALSE - (Once more, there is confusing wording here since we might not all agree on what "curving downwards" means.) A line that slopes downwards and has a curve (perhaps you call that "curving downwards " as I do) has a negative velocity (due to its negative slope). If the curve is "concave up" (you might say leveling off to a horizontal as time progresses) then the object is slowing down and the acceleration is positive.
The Meaning of Shape for a p-t Graph | The Meaning of Slope for a p-t Graph
- The slope on a velocity-time graph is representative of the acceleration of the object.
- The area on a velocity -time graph is representative of the change in position of the object.
- An accelerated object's motion will be represented by a curved line on a velocity-time graph.
- Objects with positive acceleration will be represented by upwardly-curved lines on a velocity-time graph.
- If an object is at rest, then the velocity-time graph will be a line with zero slope.
- A line with zero slope on a velocity-time graph will be representative of an object which is at rest.
- A line with a negative slope on a velocity-time graph is representative of an object with negative velocity.
- If an object changes its direction, then the line on the velocity-time graph will have a changing slope.
- An object which is slowing down is represented by a line on a velocity-time graph which is moving in the downward direction.
Answer: ABE (and almost D)
a. TRUE - Now this is important! It is the beginning point of much of our discussion of velocity-time graphs. The slope equals the acceleration.
b. TRUE - This is equally important. The area is the displacement.
c. FALSE - An object which has an acceleration will be represented by an line that has a slope. It may or may not curve, but it must have a slope other than zero.
d. FALSE - An object with positive acceleration will have an positive or upward slope on a v-t graph. It does not have to be a curved line. A curved line indicates an object that is accelerating at a changing rate of acceleration.
e. TRUE - An object that is at rest has a 0 velocity and maintains that zero velocity. The permanence of its velocity (not the fact that it is zero) gives the object a zero acceleration. and as such, the line on a v-t graph would have a slope of 0 (i.e., be horizontal).
f. FALSE - A line with zero slope is representative of an object with an acceleration of 0. It could be at rest or it could be moving at a constant velocity.
g. FALSE - A negative slope indicates a negative acceleration. The object could be moving in the positive direction and slowing down (a negative acceleration).
h. FALSE - An object which changes its direction will be represented by a line on a v-t graph that crosses over the time-axis from the + velocity region into the - velocity region.
i. FALSE - An object which is slowing down has a velocity which is approaching 0 m/s. And as such, on a v-t graph, the line must be approaching the v=0 m/s axis.
- An object that is free-falling is acted upon by the force of gravity alone.
- A falling skydiver which has reached terminal velocity is considered to be in a state of free fall.
- A ball is thrown upwards and is rising towards its peak. As it rises upwards, it is NOT considered to be in a state of free fall.
- An object in free fall experiences an acceleration which is independent of the mass of the object.
- A ball is thrown upwards, rises to its peak and eventually falls back to the original height. As the ball rises, its acceleration is upwards; as it falls, its acceleration is downwards.
- A ball is thrown upwards, rises to its peak and eventually falls back to the original height. The speed at which it is launched equals the speed at which it lands. (Assume negligible air resistance.)
- A very massive object will free fall at the same rate of acceleration as a less massive object.
- The value of g on Earth is approximately 9.8 m/s2.
- The symbol g stands for the force of gravity.
a. TRUE - Yes! This is the definition of free fall.
b. FALSE - Skydivers which are falling at terminal velocity are acted upon by large amounts of air resistance. They are experiencing more forces than the force of gravity. As such, they are NOT free-falling.
c. FALSE - Any object - whether rising, falling or moving horizontally and vertically simultaneously - can be in a state of free fall if the only force acting upon it is the force of gravity. Such objects are known as projectiles and often begin their motion while rising upwards.
d. TRUE - The unique feature of free-falling objects is that the mass of the object does not effect the trajectory characteristics. The acceleration, velocity, displacement, etc. is independent of the mass of the object.
e. FALSE - The acceleration of all free-falling objects is directed downwards. A rising object slows down due to the downward gravity force. An upward-moving object which is slowing down is said to have a downwards acceleration.
f. TRUE - If the object is truly in free-fall, then the speed of the object will be the same at all heights - whether its on the upward portion of its trajectory or the downwards portion of its trajectory. For more information, see the Projectiles page at The Physics Classroom.
g. TRUE - The acceleration of free-falling objects (referred to as the acceleration of gravity) is independent of mass. On Earth, the value is 9.8 m/s/s (the direction is down). All objects - very massive and less massive - experience this acceleration value.
h. TRUE - Yes! Know this one!
i. FALSE - Nope. A careful physics teacher will never call g the force of gravity. g is known as the acceleration of gravity. It might be best to call it the acceleration caused by gravity. When it comes to the force of gravity, we have yet another symbol for that - Fgrav. But that's a topic to be discussed in a later unit.
Introduction to Free Fall | The Acceleration of Gravity