Conceptual Questions
I. Kinematics
Velocity
During a journey over which the speed varies, the average speed in comparison to the
maximum attained speed is always
(a) one-half of it
(b) greater than it
(c) less than it
(d) equal to it
(e) none of these (Hecht 61)
Answer: (c) The average speed is defined as the total distance traveled over the total
time required to do this traveling. The maximum attained speed is an instantaneous
speed. We cannot necessarily say that the average speed is always half of the maximum
instantaneous speed. It also makes no sense that the average speed would be larger than
any speed the object had ever reached throughout the journey. The average speed would
only be equal to the maximum attained speed if the object had remained at the maximum
attained speed for the entire journey. We know that the speed varies, so this cannot be
the case. The average speed will, however, always be less than the maximum attained
speed for varying values of speed.
What two controls on a car enable a change in speed? Name another control that enables
a change in velocity. (Hewitt)
Answer: A change in speed is attained through use of the gas pedal and the brake.
Pushing the gas pedal causes the car’s speed to increase. Likewise, pushing the brake
causes the car to slow down. A change in velocity is enabled by the steering wheel. If
the car is traveling at a certain speed and someone turns the steering wheel, the car will
remain at the same speed but now be traveling in a different direction. This is a change
in velocity.
A racing automobile is traveling around a circular track at a constant speed. What is the
average velocity over one full lap? (Ballif 95)
Answer: The average velocity of an object is defined as the total displacement (NOT
distance) that the object travels in a certain amount of time. The total displacement after
one lap of the car will be zero, because it will have returned to the exact same point from
which it started. Therefore, we do not need to know the length of the track, and can
safely say that the average velocity is zero.
Acceleration
Why is it that an object can accelerate while traveling at constant speed, but not at
constant velocity? (Hewitt)
Answer: An object can accelerate while traveling at a constant speed because it can
change direction. (This is a form of acceleration – the acceleration will act in a direction
perpendicular to the original path of the object and cause it to veer off its original course).
An object cannot accelerate while traveling at a constant velocity, however, because in
order to do so, it must change either speed or direction. Since both speed and direction
are fixed by stating that an object is traveling at a constant velocity, we cannot have any
acceleration.
A car is in an intersection facing east. Describe its motion under each of the following
sets of conditions. In this question, vectors pointing east are considered positive and
vectors pointing west are considered negative.
(a) Acceleration is zero; speed is positive
(b) Acceleration is zero; speed is negative
(c) Acceleration is zero; speed is zero
(d) Acceleration is positive; speed is zero
(e) Acceleration is negative; speed is positive
(f) Acceleration is positive; speed is negative (Ballif 94)
Answer: (a) The car is traveling through the intersection, heading east, at a constant
speed.
(b) The car is backing up from the intersection at a constant speed.
(c) The car is stopped at the intersection.
(d) The car is just about to travel through the intersection after having stopped, but has
not yet actually begun to move.
(e) The car is slowing down through the intersection and still traveling east.
(f) The car is backing up from the intersection and slowing in its reverse.
A body moving with an acceleration having a constant magnitude must experience a
change in
(a) velocity
(b) speed
(c) acceleration
(d) weight
(e) none of these (Hecht 106)
Answer: (a) The weight of the body would not change. If the magnitude of the
acceleration is constant, and we don’t know the direction of the acceleration, we cannot
necessarily say that the acceleration is changing (if the direction is constant, then we have
constant acceleration; if the direction is changing, then our acceleration is not necessarily
constant). We don’t know whether the acceleration is acting parallel or perpendicular to
the body, so we can’t say with any certainty that only the speed is changing (if the
acceleration is parallel, then this is the case; if the acceleration is perpendicular, then the
direction that the body is moving in will change). Therefore, it is safe to say the
acceleration must be changing speed or direction or both, and therefore must be changing
velocity.
A mouse runs along a straight narrow tunnel. If its velocity-time curve is a straight line
parallel to the time axis, then the acceleration is
(a) a nonzero constant
(b) zero
(c) varying linearly
(d) quadratic
(e) none of these (Hecht 106)
Answer: (b) The velocity of the body is not changing in time, and therefore, it cannot be
experiencing any acceleration.
If the displacement of a body is a quadratic function of time, the body is moving with
(a) a uniform acceleration
(b) a nonconstant acceleration
(c) a uniform speed
(d) a uniform velocity
(e) none of these (Hecht 106)
Answer: (a) A quadratic displacement with respect to time would yield an everchanging slope (either increasing or decreasing). Since we know that slope corresponds
to velocity, we know that the velocity must be changing, and the body must be
undergoing acceleration. This means that the answer cannot be uniform velocity, or
uniform speed. Differentiating a quadratic yields the equation of a straight line. We
know that differentiating displacement with respect to time yields instantaneous velocity.
This means that a velocity time graph of the function would have a straight line for a
curve, corresponding to uniform acceleration.
The average speed of a coconut during a 2-s fall from a tree, starting at rest, is
(a) 19.6 m/s
(b) 9.8 m/s^2
(c) 39.2 m/s
(d) 9.8 m/s
(e) none of these (Hecht 106)
Answer: (d) We know the initial speed of the coconut to be 0 m/s. We also know that
the coconut is undergoing acceleration due to gravity (9.8 m/s^2) as it falls. The total
distance the coconut travels can be found by the equation d = v0t + 0.5at^2. We know
that the initial velocity of the coconut is 0m/s. So, we can calculate the distance to be
0.5*9.8m/s^2*(2s)^2 = 19.6m. The average speed is the total distance traveled over the
time taken, so the average speed is 19.6m/2s = 9.8m/s.
Projectile Motion
At the instant a horizontally held rifle is fired over a level range, a bullet held at the side
of the rifle is released and drops to the ground. Which bullet – the one fired down-range
or the one dropped from rest – strikes the ground first? (Hewitt)
Answer: They will strike the ground at the same time. We know we can consider the
horizontal and vertical components of velocity separately. The fired bullet will have a
horizontal velocity, but no horizontal acceleration. The dropped bullet has no horizontal
component. The vertical components, which determine the amount of time the bullet
takes to strike the ground, are the same. The dropped bullet starts with an initial velocity
of 0m/s downward, and is accelerated by gravity toward the ground at a rate of
9.81m/s^2. Likewise, the fired bullet starts with no vertical component of velocity, and
will therefore take the same amount of time as the dropped bullet to reach the ground.
II. Forces
Gravitational Force
Does an apple exert a gravitational force on the earth? How does this force compare with
the force exerted on the apple by the earth? (Ballif 171)
Answer: Yes, an apple exerts a gravitational force on the earth. This force is equal in
magnitude and opposite in direction to the force that the earth exerts on the apple. This is
a consequence of Newton’s third law. The reason that the motion of the earth does not
change noticeably due to the force acting on it as a result of the apple is that its mass is so
large. We know Newton’s second law, that acceleration equals force divided by mass.
The force on the earth is the same as the force on the apple, but the force on the earth is
divided by a substantially larger mass, resulting in an acceleration that is not even
noticeable to a person on earth.
A 1.00-kg chicken weighs 9.8N on the surface of the Earth. At a distance of one Earthradius above the planet’s surface
(a) its weight is 4.9N
(b) its mass is 0.50kg
(c) its weight is 19.6N
(d) its mass is 2.0kg
(e) none of these. (Hecht 215)
Answer: (e) None of these answers are correct. We know that the force due to gravity is
proportional to the inverse distance squared. Therefore, we know that the acceleration
due to gravity is also proportional to inverse distance squared. (This makes sense – as the
distance from the earth increases, acceleration due to gravity decreases). So, if we double
the distance from the center of the earth, we will actually reduce the acceleration due to
gravity by a factor of four, and since the mass of the chicken remains the same, we can
see that the force must also be reduced by a factor of four. So the force acting on the
chicken due to gravity is actually 2.5 N at a distance one Earth radius above the planet’s
surface.
Static Forces
If the kids on the swings are of equal weight, which swing is more likely to break? (See
diagram Hewitt p. 73)
Answer: The swing for which the supporting chains are not completely vertical is more
likely to break. (Draw a diagram resolving the forces – the tension must increase in the
ropes as the angle increases)
Frictional Forces
Which produces a greater frictional force – a wide tire or a narrow tire? Why is this?
(Hewitt)
Answer: Both will produce the same frictional force. This is because frictional force is
not dependent on surface area – rather, it depends on the coefficient of friction between
the two interacting surfaces, as well as the normal force acting on the surface. The
coefficient of friction depends on microscopic qualities of the two materials, and does not
change. The total normal force acting on the surface does not depend on its area (it will
always be a reaction force to downward forces acting on the tire), and so frictional force
does not depend upon surface area either.
Centripetal Force
A ball is whirled in a horizontal circle on the end of a string. There is a force in the string
constantly pulling the ball toward the center of the circle; yet the ball keeps going around
at the same speed. How can this fact be reconciled with Newton’s second law of motion,
which states that force causes an acceleration? (Ballif 172)
Answer: Despite the fact that the speed of the ball is constant, as it travels along in a
circle, its velocity is constantly changing and is continually tangential to the circular path.
The force acting on the string is causing an acceleration – the centripetal acceleration
which causes the ball to change direction.
A toy airplane is travelling in a circle at the end of a guide wire. It is made to go faster as
more wire is played out until both the speed and the radius of the circle are constant at
double their original values.
(a) The plane’s centripetal acceleration is unchanged.
(b) The magnitude of its centripetal acceleration is doubled.
(c) Its acceleration decreases by a factor of two.
(d) The plane’s centripetal acceleration is zero.
(e) None of these. (Hecht 214)
Answer: (b) Centripetal acceleration is defined as tangential velocity squared over the
radius of the circular path. If tangential velocity is doubled, centripetal acceleration will
increase by a factor of four. If the radius is then also doubled, the centripetal acceleration
will be cut in half, leaving a net result of twice the original centripetal acceleration.
Force and Motion
Newton’s First Law of Motion
Why would a bomb dropped from an airplane in horizontal flight remain directly under
the plane as it fell? What could prevent the bomb from remaining directly under the
plane? (Ballif 58)
Answer: The bomb dropped from an airplane will remain under the plane as it falls
because it has inertia. It will tend to maintain the motion that it had on the plane, and
therefore will have the same horizontal velocity as it had before its release (that of the
plane). It undergoes no horizontal acceleration, and therefore, it will remain under the
plane. The fact that the bomb is falling vertically will not change this, because the two
components of velocity can be considered separately. Air drag could prevent the bomb
from remaining directly under the plane.
A lawsuit developed as a result of the collision of a car and a pie truck. The truck driver
claimed that he was stopped at an intersection when the car rammed into him from
behind, causing the pies in the back of the truck to slide forward into the driver’s seat.
The car driver, on the other hand, claimed that both the truck and the car were stopped at
the intersection when then truck driver suddenly backed up and rammed into the car,
which happened to be an open convertible. The impact caused the pies to slide from the
truck into the front seat of the car. Who do you think really got hit with the pies? (Ballif
59)
Answer: The driver of the car really got hit with the pies. If the driver of the truck is
telling the truth and the car really did hit his vehicle from behind, this means that the
truck would lurch forward. However, the pies, free to move, would tend to remain in
their original position because of inertia. This means that while the truck is moving
forward, the pies are not, and they would end up sliding to the back of the truck, not the
driver’s seat as the truck driver suggested. The story of the driver of the car seems more
plausible. If the truck driver really did back into the car, the pies would tend to remain in
the motion they had previous to striking the car. This means that the pies would be
moving backwards towards the car. If the truck stopped suddenly, the pies would tend to
remain in their backwards motion because of inertia and would end up striking the front
seat of the car.
Sketch the motion of a ball released a few meters above the ground as it appears from a
reference frame that is (a) fixed to the earth (b) fixed to a flatcar in uniform motion
passing nearby. (Ballif 59)
Answer: (a) The diagram would depict the ball falling in a straight line down.
(b) The diagram would show a parabolic path.
A massive ball is suspended by a string from above, and slowly pulled by a string from
below. Is the string tension greater in the upper or the lower string? Which string is
more likely to break? Which property – mass or weight – is more important here? If the
string is instead snapped downward, which string is more likely to break? Which
property – mass or weight – is more important this time? (Hewitt)
Answer: The tension would be greater in the upper string. This is because is it balancing
out both the force of gravity (or the weight of the ball) and the tension applied by pulling
on the lower string. Therefore, the top string would be more likely to break. Weight is
the more important property here. If the bottom string were suddenly snapped
downward, it is more likely that the bottom string would break. This is because the large
mass would tend to resist motion because of its large inertia. The more important
property here is mass.
Newton’s Second Law of Motion
If we double the mass of a given object and triple the force on it, what happens to
acceleration? (Ballif 97)
Answer: The acceleration will become one and a half times greater than it was before.
If an object of mass m has an acceleration a when subject to a force F, what will be the
acceleration of an object of 3m when it is subject to the same force? (Ballif 96)
Answer: The acceleration will be one third of the old acceleration.
An object is subject to three forces: a force f to the east, a force 2f to the north, and an
unspecified force. Th object does not change its motion as a result of the forces. Draw a
diagram more or less to scale illustrating how you would find the third force, and
estimate its magnitude and direction. (Ballif 97)
Answer: In order for the object to maintain its motion, no net force must be acting on the
object. This means that the third force must have the same magnitude as the other two
forces combined but act in the opposite direction as the net direction of the other two.
The magnitude of the third force must therefore be sqrt(5)f, and it must act in the
direction (-i - 2j) – or a southwesterly direction.
A cannonball and a postage stamp are allowed to fall in a vacuum. They are released at
the same time from the same height. Describe the motion of each. What is the
acceleration of each object? Are the forces on the two objects the same as they are
falling? Why is there no contradiction between the answers to the last two questions?
Answer: In a vacuum, we can ignore the effects of air resistance. The two objects will
therefore fall straight down at the same rate. The acceleration due to gravity is the same
regardless of the mass, and would therefore act on the cannonball and the postage stamp
in the same way. The forces on the two objects are not the same as they are falling. In a
vacuum, the only force acting on each is the gravitational attraction between the objects
and the earth. The different masses of the objects mean that the forces would not be the
same even though the acceleration is. The gravitational force acting on the cannonball
would be much larger than that acting on the postage stamp. (Since acceleration is the
same for both objects, only the mass could possibly make the force different. Force is
directly proportional to mass, and so a larger mass must have a larger force on it than a
smaller mass if they undergo the same acceleration). There is no contradiction between
the last two questions because equal acceleration does not necessarily mean equal force –
the mass of the objects can change the force.
Imagine that you are standing on a cardboard box that just supports you. What would
happen to it if you jumped into the air? It would
(a) collapse
(b) be unaffected
(c) spring up as well
(d) move sidewise
(e) none of these (Hecht)
Answer: (a) In order to jump in the air, you must first exert a downward force on the box
so that the box can exert an equal but opposite force on you. Your pushing down means
that the box now needs to support your weight plus this additional force. However, if the
box was just barely supporting you before you tried to jump, this additional force on it
will cause it to collapse.
Newton’s Third Law of Motion
A man, fully dressed, is standing in the middle of a frictionless ice sheet. How can he get
off the ice? (Ballif 117)
Answer: The man can’t simply walk off the ice because it is frictionless. However, the
man knows Newton’s third law of motion – that for every action, there is an equal and
opposite reaction. Keeping this in mind, the man throws an object (a piece of clothing
perhaps) as hard as he can, realizing that his exerting a force on this object will cause the
object to exert a force on him, thereby pushing him off the ice.
Since the force that acts on a bullet when a gun is fired is equal and opposite to the force
that acts on the gun, doesn’t this imply a zero net force, and therefore the impossibility of
an accelerating bullet? Explain.
Answer: The net force acting on the system comprised of the gun and the bullet is zero.
This is consistent with Newton’s third law. However, when we are considering the
system comprised of the bullet alone, there is only one force (excluding gravity) acting
on it. This is the force which allows the bullet to have an acceleration. Likewise, the
system composed of the gun alone also has a non-zero net force acting on it. However,
because the mass of the gun is quite large compared to the mass of the bullet, this same
force acting on the gun produces a smaller acceleration on the gun.
Is it possible to devise a technique to push on a table without it pushing back on you?
(a) Yes, out in space.
(b) Yes, if someone else also pushes on it.
(c) A table never pushes in the first place.
(d) No.
(e) None of these. (Hecht 106)
Answer: (d) It is impossible to devise a technique for pushing on a table without it
pushing back on you. If you were to push on a table out in space, the table would still
push back on you, causing both you and the table to move away from your original
positions. If someone else pushes on the table with a force equal and opposite to the
force with which you push on the table, the table will not move. However, this does not
mean that the table is not pushing back on you. The table will push on the other person
with a reaction force equal in magnitude and opposite in direction to the force with which
the person pushes on the table. Likewise, the table will push back on you with a force of
equal magnitude and opposite direction.
Tension Forces
Two pictures of equal weight are hung as shown. In which arrangement is the wire most
likely to break? (See diagram Hewitt p. 73)
Answer: The wire is most likely to break in the position in which the angle between the
two supporting wires is larger. In order to support the picture, the wire must provide an
upward force equal to the weight of the picture. If the two wires are spread further apart,
this means that the tension force acting on each side of the wire must be greater in order
to provide a strong enough vertical component to balance the picture. Eventually, when
this tension force becomes too great, the wire will break.
III. Momentum
Impulse
You can’t throw a raw egg against a wall without breaking it, but you can throw it with
the same speed into a sagging sheet without breaking it. Explain. (Hewitt)
Answer: When the egg strikes something, its momentum will become zero. This change
in momentum is equal to the impulse acting on the egg. We know impulse to be force
times the time over which the force acts on the object. When the egg strikes a rigid body,
such as the wall, the time over which the force acts is quite small. This means that in
order for the impulse to equal the change in momentum, we must have a large force
acting on the egg. This is what causes the egg to break. The reason an egg can be thrown
into a sagging sheet with the same speed without breaking is also impulse. The change in
momentum in both situations is the same (both eggs have the same mass and the same
velocity and both end up with zero momentum). However, the sagging sheet increases
the time over which the impulse acts. This means that the force in this situation will be
considerably smaller than when the egg struck the wall, and the egg does not break.
A boxer being hit with a punch contrives to extend time for best results, whereas a karate
expert delivers a force in a short time for best results. Isn’t there a contradiction here?
(Hewitt)
Answer: A boxer being hit with a punch contrives to extend the time over which he is
punched so that time, rather than the force, is the more dominant quantity of impulse.
This reduces the force acting on his face. The karate expert delivers a blow in short time
to increase the force part of impulse, allowing the expert to deliver a blow to knock out
an opponent or break a block. Both people are using the same idea, that of impulse, to
achieve desired results.
A human being can survive a feet-first impact at speeds up to roughly 12m/s on concrete,
15 m/s on soil, and 34m/s on water. Explain the spread in these values.
Answer: The falling human will strike some surface and end up with no momentum.
This change in momentum corresponds to impulse. Say that a person can only withstand
a certain force acting upon him or her without being killed. On concrete, the time over
which the force acts on the person is quite small. For this reason, the force acting on the
person is quite large. In order for the person to survive, the overall impulse acting on the
person must be smaller. This means that the change in the person’s momentum must be
smaller, or they must have a smaller initial speed, to walk away from such a fall. On soil,
the time over which the force acts on the person is slightly larger, and the force acting on
the person slightly smaller. This means that the person can withstand a larger impulse
than on concrete, and can therefore survive a larger change in momentum, meaning that
the initial speed value can be slightly larger and the person can still survive. Finally, the
person landing in water has the time over which the force acts greatly extended. This
means that the force on the person will be quite a bit smaller on impact. Thus, the person
can withstand a larger change in momentum, and can therefore survive falling from a
larger speed.
Conservation of Momentum
A stationary bomb explodes into three equal masses. One travels north at a speed v; one
goes east at a speed v. In what direction and with what speed does the third one go?
(Ballif 119)
Answer: The net momentum of the two masses is sqrt(2)mv acting in a northeast
direction. We know that since momentum is conserved, the momentum of the third piece
must be equal in magnitude and opposite in direction. Therefore, the third piece must
travel at a speed of sqrt(2)v in a southwest direction.
A deuteron is a unique nuclear particle of known mass. Suppose one is accelerated to a
high speed in an atomic accelerator and directed into an observation chamber. There it
collides and sticks to a target particle that is initially at rest. As a result of the impact, the
combined target particle and deuteron is observed to move at half the initial speed of the
deuteron. Why do observers conclude that the target particle is itself a deuteron? (Hewitt
100)
Answer: Observers are able to conclude that the particle is indeed a deuteron using the
law of conservation of momentum. The initial momentum of the second particle is zero.
Therefore, we know that the initial momentum of the first particle is the same as the
momentum of the final combined particle. So, we can see that if the velocity of the
combined particle is half the original velocity, then the mass of the final particle must be
twice the mass of the deuteron, and the combined particle must be composed of two
deuterons.