Goal: Hone the concept of a collision
Source: CT151.2S02-43
A cart
of mass m is moving with speed v. Is it possible for the kinetic energy
or the momentum to be larger after a collision?
Goal: Explore momentum concepts
Source: CT151.2S02-44
A cart
of mass 2m collides and sticks to a cart of mass m that is initially at
rest. What is the speed of the combination after the collision?
(2) This question is useful for probing pre-existing ideas about
momentum, and also for distinguishing momentum ideas from kinetic
energy. It should be used just after, or even before, covering momentum
concepts.
Usually students starting momentum already have had some energy, and
kinetic energy in particular. If appropriate, they can be asked if
energy is lost in the collision. Is more or less energy lost if the
carts do not stick together?
Goal: Problem solving
Source: UMPERG-ctqpe84
A mass of 0.5 kg moving along a horizontal frictionless surface
encounters a spring having k = 200 N/m. The mass compresses the spring
by 0.1 meters before reversing its direction. Consider the total time
the mass is in contact with the spring. What is the total impulse
delivered to the mass by the spring?
(2) This problem requires students to put together the concepts
of kinetic and potential energy, and change of momentum. Some may be
tempted to resort to the definition of impulse and try to determine the
force due to the spring.
Goal: Problem solving with momentum
Source: UMPERG-ctqpe46
A
block slides along a frictionless surface and onto a slab with a rough
surface. The plot on the right shows the velocity of the blue slab as a
function of time. The slab has mass of 4kg and the block has mass of
2kg. If the block remains on top of the slab, what was its initial
speed?
(6) The block initially has velocity 3 m/s. This problem is
difficult for students. They generally have difficulty obtaining
relevant information from a diagram. In this case they must use the plot
to tell that the final velocity is 1 m/s.
Goal: Reasoning and recognizing the implications of momentum conservation.
Source: UMPERG
For ANY collision between two objects there is a time when both of the
objects are traveling with the velocity of the center of mass.
(Assume no external forces act on either object.)
(2) This statement is false despite the fact that it is true for just
about all of the instances of collision that students see. In a
perfectly inelastic collision it is certainly true that both bodies have
the velocity of the center of mass after the collision. In a general one
dimensional collision with only spring forces it is also true. For the
statement to be true about a specific collision, there must be a time
when the relative velocity of the two objects is zero. The statement is
clearly false in general for two-dimensional collisons. As an example of
a one-dimensional collision for which the statement is false, consider a
bullet that passes through a block of wood initially at rest. The bullet
slows down and the block speeds up but they never have the same
velocity.
Goal: Interrelate and contrast the concepts of work, kinetic energy and impulse.
Source: UMPERG-ctqpe96
Compare two collisions that are perfectly inelastic. In case (A) a car
traveling with velocity V collides head-on with a sports car having half
the mass and traveling in the opposite direction with twice the speed.
In case (B) a car traveling with velocity V collides head-on with a
light truck having twice the mass and traveling in the opposite
direction with half the speed. In which case is the work done on the
car during the collision the greatest?
(4) The total momentum of both systems is zero, so after the collision
there is no KE in either system. System (A) has more kinetic energy
initially. There is no way, however, to determine how much of the
kinetic energy in the combined system of the two vehicles is dissipated
in the automobile as opposed to the other vehicle.
This question serves only to provoke a discussion of the dissipation of
energy in a collision. Students are tempted to assume that each
vehicle must absorb its own initial KE.
How do the forces acting on the car in the two cases compare?
Which collision takes longer?
Which vehicle do you think will suffer the greatest damage?
Promote a discussion of auto safety.
Goal: Hone the concept of impulse and recognize an application of the 3rd law.
Source: UMPERG-ctqpe92
Compare two collisions that are perfectly inelastic. In case (A) a car
traveling with velocity V collides head-on with a sports car having half
the mass and traveling in the opposite direction with twice the speed.
In case (B) a car traveling with velocity V collides head-on with a
light truck having twice the mass and traveling in the opposite
direction with half the speed. In which case is the impulse delivered
to the car during the collision the greatest?
(3) The impulse delivered to the automobile is the same in both cases.
In both cases the initial momentum of the automobile is MV to the right
and the final momentum is zero.
Impulse is related to the change in momentum. This question provided the
opportunity to discuss the definition of impulse [integral of force over
time interval] and its relation to momentum change. Many students think
I=Δp is the definition of impulse rather than the result of Newton’s
second law. Students should realize that no statement can be made about
the forces exerted on the two cars – only that the integral of the force
over the collision time is the same.
How do the forces acting on the car in the two cases compare? Which
collision takes longer?
Set up the comparison with collision carts.
Goal: Problem solving using momentum conservation.
Source: UMPERG-ctqpe90
On an icy road, an automobile traveling east with speed 50 mph collides
head-on with a sports car of half the mass traveling west with speed 60
mph. If the vehicles remain locked together, the final speed is:
(8) None of the above. This is a straightforward totally inelastic
collision situation.
Students are frequently bothered by the idea of a totally or perfectly
inelastic collision. They are inclined to think of inelasticity as
imperfection, so the idea of perfect imperfection is distressing.
Consequently the scale shifts and they label collisions when objects
stick together as inelastic, the general collision as elastic, and
collisions conserving kinetic energy as perfectly elastic.
How fast would the car have to be traveling for the combined vehicles to
remain at rest after the collision?
If the collision was elastic, in which direction would the sports car
travel after the collision?
By relating the general collision problem to that of two masses
colliding with a spring between them, it is possible to get students to
realize that all two body collisions pass through the state with both
objects traveling with the CM velocity. This helps unify the concepts
of elastic, inelastic and perfectly inelastic collisions.
Goal: Hone the vector nature of impulse and contrast impulse to kinetic energy.
Source: UMPERG-ctqpe82
A block having mass M travels along a horizontal frictionless surface
with speed vx. What impulse must be delivered to the mass
to reverse its direction?
(1) or (2) or (6) are all defensible answers depending upon how students
interpret the question. This is a good question for stressing to
students that it is their reasoning not their answer that is important.
Impulse is a vector. Using the impulse-momentum relation, the change in
momentum must be at least mvx in the direction opposite to
the motion to reverse direction.
What impulse would be needed to make the mass travel parallel to the
y-axis?
Suppose a constant force Fx acts for 4 seconds causing the
mass to stop. What force would be needed to stop the mass in 2 seconds.
Have students make a concept map showing the relationships among the
quantities mass, velocity, momentum, impulse, time, force, and average
force.
Does it bother students that, in 10 seconds gravitation provides an
impulse of 10mg to a book whether it is dropped or sitting on a table?
Commentary:
Answer
(4) Many students get fixated on the idea that all collisions are head
on and cause the object to lose energy and momentum.