Vectors and Projectiles
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Part A: Multiple-Multiple Choice
1. Which of the following statements are true of scalars and vectors? List all that are TRUE.
- A vector quantity always has a direction associated with it.
- A scalar quantity can have a direction associated with it.
- Vectors can be added together; scalar quantities cannot.
- Vectors can be represented by an arrow on a scaled diagram; the length of the arrow represents the vector's magnitude and the direction it points represents the vector's direction.
Answer: AD
a. TRUE - Vectors are defined as quantities which are fully described by both their magnitude and direction. By definition, a vector has a direction associated with it. If it didn't, then it would NOT be a vector.
b. FALSE - Scalars are defined as quantities which are fully described by their magnitude alone. Scalars have no regard for direction and it is meaningless to associate a direction with such a quantity. If a quantity did have a direction associated with it, then that quantity would not be a vector.
c. FALSE - Both vectors and scalars can be added together. The rules for adding vectors together are unique to vectors and cannot be used when adding scalars together. The direction of a vector must be considered when adding two vectors together. Direction is of no importance when adding scalars.
d. TRUE - This is exactly the case and exactly what is done throughout the unit.
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2. Which of the following quantities are vectors? Include all that apply.
- distance traveled
- displacement
- average speed
- average velocity
- instantaneous velocity
- acceleration
Answer: BDEF
Of the five kinematic quantities listed here (distance, displacement, speed, velocity and acceleration), three of them are vectors. Displacement, velocity (both average and instantaneous), and acceleration all require the mention of a direction in order to fully describe the quantity.
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3. Numerical values and directions are stated for a variety of quantities. Which of these statements represent a vector description? Include all that apply.
- 20 meters, west
- 9.8 m/s/s
- 35 mi/hr, south
- 16 years old
- 60 minutes
- 3.5 m/s/s, south
- -3.5 m/s/s
- +20 degrees C
Answer: ACFG
Expressions of vector quantities would include a magnitude (number, value, etc.) and a direction. The direction could be described as being north, south, east, west or left, right, up, down. On occasion, a "+" or "-" is used to describe the direction. Since mathematical computations on calculators do not fare well with the typing of "south," a - sign is often substituted for a given direction. In the case of g, the units indicate an acceleration quantity. The "-" sign indicates a direction. One must be careful in assuming that a "+" or "-" sign is a sure sign of a quantity being a direction for other non-vector quantities can use such signs as well (as is the case in h).
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4. Which of the following statements are true of vector addition, vector subtraction, and vector addition diagrams? List all that apply.
- Vectors A, B, and C are added together as A + B + C. If the order in which they are added is changed to C + B + A, then the result would be different.
- Vectors A, B, and C are added together as A + B + C. If the order in which they are added is reversed to C + B + A, then the result would be a vector with the same magnitude but the opposite direction.
- When constructing a vector diagram for A + B + C, it is not absolutely necessary that vectors B and C use the same scale that is used by vector A.
- The resultant in a vector addition diagram always extends from the head of the last vector to the tail of the first vector.
- If vectors A and B are added at right angles to each other, then one can be sure that the resultant will have a magnitude that is greater than the magnitudes of either one of the individual vectors A and B.
- If vectors A and B are added at right angles to each other, then one can be sure that the resultant will have a magnitude that is less than the arithmetic sum of the magnitudes of A and B.
- Vector addition diagrams cannot be used to determine the resultant when there is a vector subtraction operation.
Answer: EF
a. FALSE - Altering the order in which three vectors are added does not alter the result of the addition process. A + B + C = C + B + A. Each order of operation yields a resultant with the same magnitude and direction.
b. FALSE - As mentioned above in a, altering the order in which three vectors are added does not alter the result of the addition process. Reversing the order produces a resultant with the same magnitude and the same direction.
c. FALSE - When constructing a vector addition diagram, a scale must be chosen and adhered to. The scale which used to draw vector A must also be used for vectors B and C. One cannot switch horses in the middle of the stream.
d. FALSE - The resultant in a vector addition diagram is drawn from the tail of the first vector (the starting point) to the head of the last vector (the finishing point).
e. TRUE - Suppose that A = 3 units and B = 4 units and that the two vectors are directed at right angles to each other. The vector sum or resultant of A + B is 5 units, which is clearly greater than either one of the vectors being added. In general, the resultant in such a case will be represented on a vector addition diagram as the hypotenuse of a right triangle. The hypotenuse is always greater than the other two legs of the triangle. So this statement is always true.
f. TRUE - Suppose that A = 3 units and B = 4 units and that the two vectors are directed at right angles to each other. The vector sum or resultant of A + B is 5 units whereas the arithmetic sum is 7 units. In this case and in all cases, the vector sum of two right angle vectors will always be less than the arithmetic sum. That is, Sqrt(a^{2} + b^{2}) will always be less than a + b.
g. FALSE - When a vector subtraction operation is performed, it is usually advisable to simply convert it into a vector addition operation. This is accomplished by adding the negative of the vector which is being subtracted. So A - B would be equivalent to A + (-B). By so doing, a vector addition diagram can be used to determine the resultant.
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5. Which of the following descriptions of moving objects accurately portray a projectile? List all that apply.
- an object which is moving through the air and not touching any surface
- a falling skydiver with an open parachute
- any object upon which air resistance is negligible
- a free-falling object
- an object upon which the only significant force is the force of gravity
- a falling feather
- a falling feather in a vacuum chamber
- a falling feather in a falling vacuum chamber.
Answer: DEGH
A projectile is an object upon which the only force is gravity. Air resistance must be negligible or nonexistent. Other forces resulting from people or things pulling or pushing, attached strings or contact with surfaces must not be present.
a. NO - A plane moves through the air and is not touching any surface. Yet, a plane is clearly not a projectile.
b. NO - A falling skydiver typically experiences considerable air resistance. It is popular to describe such skydivers as being in free fall. This is an erroneous use of the term.
c. NO - As you sit in your chair, air resistance is negligible. You are certainly not a projectile (at least, we hope not).
d. YES - A projectile is an object in free fall.
e. YES - An object upon which the only significant force is gravity fits the definition of a projectile (provided that significant means "having an influence").
f. NO - Falling feathers encounter air resistance which impedes the downward acceleration and causes the feather to fall at nearly a constant velocity.
g. YES - When a feather is allowed to fall in a vacuum, air resistance is eliminated and the feather can free fall.
h. YES - When a feather is allowed to fall in a vacuum and the vacuum is free-falling as well, air resistance is eliminated and an observer would notice that both the vacuum chamber and the feather are in free fall.
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6. Which of the following statements are true of projectiles? List all that apply.
- A projectile is a free-falling object.
- A projectile experiences negligible or no air resistance.
- A projectile must be moving in the downward direction.
- A projectile must be accelerating in the downward direction.
- A projectile does not have to have horizontal motion.
- A projectile could begin its projectile motion with a downward velocity.
- A projectile does not need to be "falling."
Answer: ABDEF and possibly G
a. TRUE - Free-falling objects, like projectiles, are objects upon which the only significant force is gravity.
b. TRUE - The only force on a projectile is gravity; air resistance must not be present or must not have an influence upon the motion of the projectile.
c. FALSE - Projectiles can be moving either upward or downward or at an angle to the vertical. They must however be accelerating downward, consistent with gravity's effect on an object.
d. TRUE - The force of gravity acts directly downwards upon an object, causing a downward acceleration. Any projectile must be accelerating downwards regardless of other features of its motion.
e. TRUE - A projectile could be moving strictly in a vertical direction with no horizontal motion. A ball thrown straight up in the air would be such a case.
f. TRUE - There is no rule about which direction a projectile must be moving at the instant it is projected. It could begin its motion with a initial downward velocity.
g. TRUE - The word "falling" can mean different things to different people. If "falling" involves moving in the downward direction at all instants in time, then a projectile does not need to be "falling." To many, "falling" means being pulled downward by gravity's force. In this case, a projectile must be "falling."
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7. Which of the following statements are true of the horizontal motion of projectiles? List all that apply.
- A projectile does not have a horizontal velocity.
- A projectile with a rightward component of motion will have a rightward component of acceleration.
- The horizontal velocity of a projectile changes by 9.8 m/s each second.
- A projectile with a horizontal component of motion will have a constant horizontal velocity.
- The horizontal velocity of a projectile is 0 m/s at the peak of its trajectory.
- The horizontal velocity of a projectile is unaffected by the vertical velocity; these two components of motion are independent of each other.
- The horizontal displacement of a projectile is dependent upon the time of flight and the initial horizontal velocity.
- The final horizontal velocity of a projectile is always equal to the initial horizontal velocity.
- As a projectile rises towards the peak of its trajectory, the horizontal velocity will decrease; as it falls from the peak of its trajectory, its horizontal velocity will decrease.
- Consider a projectile launched from ground level at a fixed launch speed and a variable angle and landing at ground level. The horizontal displacement (i.e., the range) of the projectile will always increase as the angle of launch is increased from 0 degrees to 90 degrees.
- Consider a projectile launched from ground level at a fixed launch angle and a variable launch speed and landing at ground level. The horizontal displacement (i.e., the range) of the projectile will always increase as the launch speed is increased.
Answer: DFGHK
a. FALSE - Many projectiles are moving from left to right and from right to left as they simultaneously free fall. Such projectiles have a horizontal motion. While a projectile can have a horizontal motion, it cannot have a horizontal acceleration. Whatever motion which it has in the horizontal dimension, must be motion with a constant velocity.
b. FALSE - A projectile with a rightward motion (in addition to a vertical motion) will have a constant velocity in the rightward direction. This is to say that it has no horizontal acceleration.
c. FALSE - A projectile has a constant horizontal velocity. The vertical velocity will change by 9.8 m/s each second.
d. TRUE - Absolutely true! Projectiles are objects being acted upon by gravity alone. As such, there is a vertical acceleration but no horizontal acceleration. The horizontal velocity of a projectile is either zero or a constant nonzero value.
e. FALSE - The vertical velocity of a projectile is 0 m/s at the peak of its trajectory; but the horizontal component of the velocity at the peak is whatever the value was when first launched.
f. TRUE - For any two dimensional motion (whether projectile motion or riverboat problems or ...), perpendicular components of the motion are independent of each other. Any alteration in a vertical component will not effect the horizontal components of motion.
g. TRUE - The horizontal displacement (x) can be calculated with the formula x = v_{ox} • t, where v_{ox} is the initial horizontal velocity and t is the time. These are the two variables which effect the horizontal displacement of a projectile.
h. TRUE - Since there is no horizontal acceleration for a projectile, the initial horizontal velocity is equal to the final horizontal velocity.
i. FALSE - This is a true description for the vertical component of the velocity. The horizontal velocity is unchanging throughout the trajectory of a projectile.
j. FALSE - The range (or horizontal displacement) will increase as the angle is increased from 0 degrees to 45 degrees. The maximum range occurs at 45 degrees. As the angle is further increased to values greater than 45 degrees, the horizontal displacement decreases.
k. TRUE - As the launch speed is increased, the components of the initial velocity (both the horizontal and the vertical) increase as well. This causes the projectile to stay in the air for a longer period of time and to be moving faster in the horizontal direction. The result is that increased launch speeds always lead to increased horizontal displacements.
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8. Which of the following statements are true of the vertical motion of projectiles? List all that apply.
- The vertical component of a projectile's velocity is a constant value of 9.8 m/s.
- The vertical component of a projectile's velocity is constant.
- The vertical component of a projectile's velocity is changing.
- The vertical component of a projectile's velocity is changing at a constant rate.
- A projectile with an upward component of motion will have a upward component of acceleration.
- A projectile with an downward component of motion will have a downward component of acceleration.
- The magnitude of the vertical velocity of a projectile changes by 9.8 m/s each second.
- The vertical velocity of a projectile is 0 m/s at the peak of its trajectory.
- The vertical velocity of a projectile is unaffected by the horizontal velocity; these two components of motion are independent of each other.
- The final vertical velocity of a projectile is always equal to the initial vertical velocity.
- The vertical acceleration of a projectile is 0 m/s/s when it is at the peak of its trajectory.
- As a projectile rises towards the peak of its trajectory, the vertical acceleration will decrease; as it falls from the peak of its trajectory, its vertical acceleration will decrease.
- As a projectile rises towards the peak of its trajectory, the vertical acceleration is directed upward; as it falls from the peak of its trajectory, its vertical acceleration is directed downward.
- The peak height to which a projectile rises above the launch location is dependent upon the initial vertical velocity.
- As a projectile rises towards the peak of its trajectory, the vertical velocity will decrease; as it falls from the peak of its trajectory, its vertical velocity will decrease.
- Consider a projectile launched from ground level at a fixed launch speed and a variable angle and landing at ground level. The vertical displacement of the projectile during the first half of its trajectory (i.e., the peak height) will always increase as the angle of launch is increased from 0 degrees to 90 degrees.
- Consider a projectile launched from ground level at a fixed launch angle and a variable launch speed and landing at ground level. The vertical displacement of the projectile during the first half of its trajectory (i.e., the peak height) will always increase as the launch speed is increased.
Answer: CDFGHINPQ
a. FALSE - The vertical component of a projectile's velocity is constantly changing. It is the acceleration which has a value of 9.8 m/s/s.
b. FALSE - Projectiles are objects being acted upon by gravity alone. As such, there is a vertical acceleration; the vertical velocity is not constant, but changing.
c. TRUE - See part b above.
d. TRUE - A projectile has a vertical acceleration of 9.8 m/s/s throughout the entire trajectory. This acceleration value is constant. This means that the vertical velocity changes by the same amount - 9.8 m/s - during each second of its motion. There is a change in the vertical velocity by a constant amount.
e. FALSE - All projectiles experience a downward acceleration, whether they are moving upward or downward. The upward-moving projectiles have an upward velocity, but the actual velocity values are getting smaller; that is, the projectile is slowing down on the way to its peak.
f. TRUE - This is a true statement. It could also be said that a projectile with an upward component of motion also has a downward acceleration. All projectiles accelerate in the downward direction. Period.
g. TRUE - This is absolutely true .
h. TRUE - At the peak of its trajectory, a projectile is in the process of changing directions. The vertical velocity must change from a positive value (+ for upward) to a negative value (- for downward). This transition means that the value for the vertical velocity must at sometime be in between a + and - number. The in-between number is 0 m/s and this occurs at the peak.
i. TRUE - For any two dimensional motion (whether projectile motion or riverboat problems or ...), perpendicular components of the motion are independent of each other. Any alteration in a vertical component will not effect the horizontal components of motion.
j. FALSE - A projectile launched at an angle forms a parabolic trajectory. Suppose that one were to trace a projectile's motion forward in time from the peak and backwards in time from the peak. If done, one would find that the vertical velocity value has the same magnitude for equal amounts of times traced forward and backward from the peak. So for the same time before and after the peak, a projectile has the same speed. However, some projectiles are not launched from the same height at which they land. The final height is not the same as the initial height and as such the time to rise to the peak is not equal to the time to fall from the peak. In such instances, the initial vertical velocity is not equal to the final vertical velocity.
k. FALSE - No! No! No! The vertical velocity is 0 m/s at the peak and the vertical acceleration is -9.8 m/s/s throughout the entire trajectory.
l. FALSE - This would be a true description of the vertical velocity. But the vertical acceleration is a constant value of 9.8 m/s/s throughout the entire trajectory.
m. FALSE - Not only is the magnitude of the vertical acceleration a constant value throughout a projectile's trajectory, the direction is constant as well. Projectile's at all times regardless of any other variable will accelerate downwards at 9.8 m/s/s. This is perhaps the most important truth to digest about projectiles.
n. TRUE - The initial vertical velocity has an effect on the time taken by a projectile to rise towards its peak. It also effects the average speed of the projectile as it rises towards its peak. As a result, any alteration in the vertical velocity will alter the peak height of the projectile.
o. FALSE - Upward-rising projectiles have a downward acceleration; this means they are slowing down as they rise. The magnitude of their velocity is decreasing. Downward-moving projectiles also have a downward acceleration; this means they are speeding up. The magnitude of their velocity is increasing.
p. TRUE - An increase in the angle of launch (from 0 to 90 degrees) will always increase the vertical component of the initial velocity (v_{iy}). This increase in v_{iy} will lead to increased times for the projectile rising towards its peak. And an increased angle causes the projectile to move with a greater average speed during its path towards its peak. Both of these effects lead to the outcome that the peak height of a projectile will increase as the angle of launch increases from 0 to 90 degrees.
q. TRUE - As the launch speed is increased, the components of the initial velocity (both the horizontal and the vertical) increase as well. This causes the projectile to stay in the air for a longer period of time and to be moving faster in the vertical direction. The result is that increased launch speeds always lead to increased heights for projectiles.
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9. Which of the following statements are true of the time of flight for a projectile? List all that apply.
- The time that a projectile is in the air is dependent upon the horizontal component of the initial velocity.
- The time that a projectile is in the air is dependent upon the vertical component of the initial velocity.
- For a projectile which lands at the same height that it is projected from, the time to rise to the peak is equal to the time to fall from its peak to the original height.
- For the same upward launch angles, projectiles will stay in the air longer if the initial velocity is increased.
- Assume that a kicked ball in football is a projectile. If the ball takes 3 seconds to rise to the peak of its trajectory, then it will take 6 seconds to fall from the peak of its trajectory to the ground.
Answer: BCD
a. FALSE - The time for a projectile to rise vertically to its peak (and subsequently fall back to the ground) is dependent upon the initial vertical velocity. Alteration in the horizontal velocity will only cause the projectile to have a greater horizontal displacement (x).
b. TRUE - Absolutely true. Projectiles with a greater vertical component of initial velocity will be in the air for longer amount of times (assuming that the direction of v_{iy} is upward). An alteration in the v_{iy} value will alter the time of flight of the projectile, regardless of the direction of v_{iy}.
c. TRUE - For projectiles launched at upward angles and landing at the original height, the time to the rise to the peak equals the time to fall from the peak. If it takes 3 seconds to rise upward, it will take 3 seconds to fall.
d. TRUE - For a constant launch angle, an increase in the initial velocity (v_{i}) will increase the vertical velocity (v_{iy}). This results in an increased time for the projectile to decelerate to 0 m/s as it rises towards its peak. So the projectile takes longer to get to the peak, longer to fall from the peak and overall is in the air for a longer time.
e. FALSE - Close, but very false. If it takes 3 seconds to rise to the peak, then it takes 3 seconds to fall from the peak; The 6 seconds is the total time of flight of the projectile.
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Navigate to:
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Questions and Links
Answers to Questions:
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#56-72
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