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Force and Motion Unit Overview |
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The Force and Motion Unit is intended to
engage students in myriad experiences with hands-on and computer-based materials
that will help them modify their existing ideas and construct new ideas about
the motion, forces, and their relation to each other. Cycle I and Cycle II use the Microcomputer
Based Laboratory and simulators to help students represent unidirectional
motion. Cycle II helps students to establish relationships between force and
motion and Cycle III focuses on bi-directional motion. In Cycle IV students use bathroom scales,
spring scales, simulators, and the MBL to get a sense of equal and opposite forces
and conservation of momentum. Cycle
V focuses on gravity and also on balanced forces and inertia in the vertical
direction. Cycle VI is most
appropriate for high school physics courses focusing on two-dimensional and
rotational motion. The focus of Cycle VII is on energy transfer and energy
conservation. The seven cycles in this unit are as follows: Cycle
I: Representing One Dimensional Motion in One Direction Cycle
II: Connecting Forces and Motion in One Direction Cycle
III: Changing Direction Cycle
IV: Impulse and Momentum Cycle
V: Gravity Cycle
VI: Motion in Two Dimensions Cycle VII: Energy Considerations (now available for download) Because this unit is computer intensive
(including many simulator and MBL activities) there is no paper and pencil
version. However, it is possible to
use this unit in a class that has only a few computers. Students can share two or three computer
“stations” where the MBL or simulator activities are set up ahead of time. If a printer is connected to the computer,
students can then easily print out the results of their experiments and put
them in their notebooks. Also, the
computer version of the activities is designed for students to edit each
document (type in predictions, explanations, and pasting pictures)
electronically so only a small amount of space is provided for answers. If you intend to print these activities
out to distribute to students you may wish to alter them so that more space
is provided for predictions and explanations. Target Ideas for
Unit
The Force and Motion Unit was designed to
provide opportunities for students to construct ideas that are closely
aligned with the ones listed below.
At the end of each activity in the development phase, students are
asked to add or modify an idea in their Idea Journal, based on evidence
gathered within that activity. We
have found this semi-structured approach for development of a common set of
ideas to work well with high school students, prospective and in service
elementary teachers. Naturally, as
part of their consensus discussion for each cycle, students will probably
develop these ideas in their own words.
However, the conceptual content of their own ideas should be similar
to these. The Teacher Guide for each cycle provides examples of the kinds of
statements students actually develop in the class. After the class agrees on
a set of ideas the teacher should introduce appropriate terminology and
conventions so that the students' are more closely aligned with the
corresponding ideas they would find in textbooks or when they talk with other
students. Target Ideas for Cycle I
1. Distance-Time
Graph Idea: An object moving at a constant
speed has a distance-time graph that is a straight line with a positive
slope. The speed of an object can be determined
by finding the slope of the distance-time graph. Objects that are moving at faster constant speeds have steeper
slopes. Objects moving with
non-constant speed have distance-time graphs that are curved. If the object is speeding up, the curve is
concave upward. If the object is slowing down, the curve is concave
downward. Objects that are not moving
have a distance-time graph that is a horizontal straight line. For a curved distance-time graph, the
slope of the tangent line at any point gives the instantaneous speed. The average speed is found by taking the
slope of the line segment joining the two points on the graph that
corresponds to the time interval. 2. Speed-Time Graph Idea: An object moving at a constant speed has a speed-time graph that is a horizontal straight line. Faster speeds have higher values on the speed axis and slower speeds have lower values on the speed axis. An object that is increasing its speed at a constant rate has a speed-time graph that is a straight line with a positive slope. An object that is decreasing its speed at a constant rate has a speed-time graph that is a straight line with a negative slope. The speed-time graph can be determined from a distance-time graph by finding the slope of the distance-time graph. 3. Strobe Diagram Idea: Objects that are moving at constant speed have dots that are evenly spaced. Objects that are increasing their speed (accelerating) have dots that are increasingly further and further apart. Objects that are decreasing their speed have dots that become closer and closer together.
4. Speed,
Change-in-Distance, Change-In-Time Equation Idea: The speed is equal to the
distance traveled divided by the time interval.
5. Acceleration-Time
Graph Idea: An object speeding up at a
constant rate has an acceleration-time graph that is a horizontal line with a
positive value. A faster rate of
acceleration has a higher value, a slower rate has a lower value. An object that is slowing down has an
acceleration-time graph that is a horizontal line with a negative value. An object that is moving at constant speed
has zero acceleration, so the acceleration-time graph is a horizontal
straight line at zero. The
acceleration of an object can be found by taking the slope of a speed-time
graph. 6. Acceleration,
Change-In-Speed, Change-In-Time Equation Idea The acceleration is equal to the
change in speed divided by the change in time.
Target Ideas for
Cycle II 1.
Friction Idea: When
an object moves on a surface, there is a friction force acting on the object
that is opposite to the direction of its motion. The strength of the friction force depends on the nature of the
two touching surfaces. Generally, the
rougher the surfaces the stronger the friction force, the smoother the
surfaces, the weaker the friction force.
The stronger the friction force, the more quickly an object will slow
down.
Either This is also known
as Newton’s Second Law of Motion
Target Ideas for Cycle III
1.
Velocity versus Speed Idea: The speed of an object is a quantity that tells only how fast the
object is moving. The larger the
number, the faster the object is moving.
The velocity of an object is a different quantity that tells both the speed and direction of the object’s motion. The size of the velocity number tells how fast the object
is moving (its speed). The sign of the velocity tells the direction the object is
moving. A common method of
identifying the direction an object is moving is to make the velocity positive
when the object is moving to the right (or away from the motion detector) and
negative when the object is moving to the left (or towards the motion
detector).
2.
Position
versus Distance Idea: Distance is a quantity that tells only how
far an object has traveled. Distance
is always positive and does not indicate in which direction an object is
moving. Position, on the other hand,
can be positive or negative. The
position of an object gives its location with respect to some reference
point. A common method of identifying
the location of an object is to make the sign of the position positive when
the object is to the right of the reference point, and negative when the
object is to the left of the reference point. The magnitude of an
object’s position indicates how far away the object is from the reference
point, and the sign of the
position indicates in what direction. The distance-time
and position-time graphs below are for two objects, one (green) moving to the
left and the other (blue) moving to the right.
3.
Displacement Idea:
The displacement of an object between two
instants of time is defined as the position of the object at the later time
minus its position at the earlier time.
In symbols, if x1 is its position at time t1,
and x2 is its position at time t2, then the
displacement
Another way of expressing this is, the displacement of an object is equal to its change in position, which can be found be taking is final position minus its initial position. This can be written as
4. Instantaneous Velocity versus Average Velocity Idea: The average velocity of an object between two instants of time is defined as the displacement divided by the time interval. The time interval is the later instant of time minus the earlier instant of time. In symbols:
The value of the instantaneous velocity at an instant of time is equal to the slope of the tangent line to the position-time graph at that time. The value of the instantaneous velocity can also be determined from the ordinate value (vertical axis value) on the velocity-time graph. 5. Sign of Acceleration Idea: Average acceleration equals the change in velocity divided by the time interval. If the object moves with constant acceleration, the average acceleration is equal to the acceleration at each instant of time. The average acceleration also equals the slope of the velocity-time graph between two points. In symbols:
Positive acceleration either means speeding up in the original direction (to the right), or slowing down in the opposite direction (to the left). Negative acceleration either means slowing down in the original direction (to the right), or speeding up in the opposite direction (to the left).
6. Non-Constant Force Idea: Newton’s Second Law of motion is true for non-constant forces as well as for constant forces. When the force changes in direction and magnitude, the acceleration is always acting in the same direction as the force at the same time. The force is directly proportional to the acceleration.
NOTE: The area under the velocity-time and
acceleration-time graphs was not addressed during the development
activities. Instructors may wish to
address this issue at this time by having students use the accumulator tool
in the simulator to compare the area under a velocity-time graph with the
position-time graph to show that this gives the displacement, and the area under an acceleration-time
graph gives the velocity. Target Ideas for Cycle IV
1.
Comparative Strengths of Forces of Interacting
Objects Idea: All forces involve
interactions between two or more objects.
When two objects are interacting, by pushing or pulling on each other,
whether at rest or in motion, the pushing or pulling forces are equal in
strength but opposite indirection. Two interacting objects exert equal forces on each
other despite the mass difference, size difference or the motion of one
object or the other. This is actually Newton’s Third Law of Motion. Another way to state it is: When two objects interact, the strength of the
force that one object exerts on the other is equal to the strength of the
force that the second object exerts on the first, but in the opposite
direction. 2.
Unbalanced
Force, Duration, and Change In Momentum Idea: The
momentum of an object is its mass times its velocity. When an unbalanced force acts on an object
over time, its momentum will change.
The product of the unbalanced force and its duration equals change in
momentum of the object. When an unbalanced force acts on an object, (a) the longer the force acts on the object, the greater the change in the object’s velocity. (b) the greater the mass of the object, the smaller the change in the object’s velocity. (c) the change in the object’s velocity causes a change in the object’s momentum. (d) the change in velocity of the object does not depend on the initial velocity of the object. Momentum is defined as the product of an object’s mass and velocity. The symbol for momentum is p, so this can be written as Momentum = mass x velocity
Change in Momentum = (mass) x (change in velocity) 3.
Change
in Momentum During Interactions Idea: When two objects
interact, the momentum of each is changed by the same amount. In other words, when two objects interact, the change in momentum of the first object is equal to the change in momentum of the second object regardless of the masses of the objects or their relative velocities.
This is known as the
Law of Conservation of Momentum. 4.
Representing
Impulse Idea: Force generally acts over a time
interval. The product of the force
and the time interval over which the force acts is called the impulse.
If an impulse is applied to an object, then the object’s velocity will change, and thus cause a change in momentum. The impulse is equal to the change in momentum of an object.
The units of momentum (kg m/s) and impulse (N s) are equivalent.
The area under a force-time graph is equal to the impulse. 5.
Total
Momentum During Interactions Idea: When two objects
interact, the sum of the individual momentum of the two objects before the
interaction is equal to the sum of the individual momentum of the two objects
after the interaction.
This is true regardless of the masses of the object, whether they stick together after the interaction (inelastic collision), they bounce off each other (elastic collision), or whether one or both objects are moving before the interaction. Target Ideas for
Cycle V 1.
Magnetic and Electric Forces Idea: The magnetic and electrical interactions are examples
of forces that can act on an object at a distance, without contact. Magnetic forces act only on objects that contain
iron. Electric forces seem to attract all objects,
regardless of the kind of material The strength of the magnetic force on an object
increases with the amount of magnetic stuff (iron) contained in the
object. The stronger the magnet, the
stronger the magnetic force. The strength of the electric force can be
increased by rubbing harder or longer.
The amount of material does not matter. A magnet exerts an attractive force on
non-magnetized objects that contain iron.
A rubbed electric object exerts an attractive force on non-rubbed
objects. 2.
The gravitational force idea: The gravitational interaction, which also acts at a distance, occurs
between all objects that have mass.
There is a gravitational force between the Earth and all objects on
the Earth. The gravitational force is
an attractive force. The direction of
this force is downward, toward the center of the Earth, and its strength is
proportional to the amount of gravitational stuff (mass) contained in the
object.
Heavier objects
resist a change in their motion more than lighter objects, so it takes a
greater force to accelerate a heavier object at the same rate as a lighter
object. Two objects of
different mass fall to the ground and hit at the same time. They fall at the same rate of 9.80
m/s/s. The force on the heavier
object is greater than the force on the lighter object. An object with a
larger mass has more inertia (or resistance to a change in its state of
motion), which makes it necessary to apply a greater force to accelerate if
at the same rate as an object whose mass is smaller. Big mass č
Big Inertia č More Force Needed Small mass č
Small Inertia č
Less Force Needed
To have the same
acceleration, if m is bigger, F must be bigger, if m is smaller, F must be
smaller.
In space, far away
from the Earth or other bodies, where the gravitational force on an object
would be zero, the object has zero weight, but it still has the same amount
of mass as on the Earth. The weight of an
object is due to the gravitational force acting on it.
When an object is
resting motionless on a table, there are two forces acting on it, the
gravitational force downward and the force of the table going upward. The force exerted upward by the table is
called the normal force. The normal force always acts on an object
perpendicular to the surface on which it is resting.
Target Ideas for Cycle VI
1.
Combining Velocities Idea: When an object receives a succession of impulses that are equal in
magnitude and duration, the velocity of the object will increase at a
constant rate. Each impulse will give
the object a certain change in velocity.
Each succeeding impulse will change the velocity by the same amount,
thus the change in velocities add together.
Example:
If one impulse increases an object’s velocity by 3 m/s, then after a
second impulse of the same magnitude, the object’s velocity will have
increased to 6 m/s. After a their
impulse, the velocity will be 9 m/s, etc. If an object is moving in one direction at a
constant rate and an impulse is applied at right angles to its motion, the
two velocities combine together to cause a change in the object’s
velocity. The magnitude of the
resultant velocity can be found by vector addition.
2.
Force
Direction and Change in Velocity Idea: When a force
is applied in the direction an object is moving, the object’s velocity will
increase and it will continue moving in the same direction. When a force is
applied in the opposite direction to an object’s motion, the object will slow
down. When a force is
applied at right angles to the direction of motion of an object the direction
of the object’s motion will change, but the magnitude of its velocity (its
speed) will remain constant. (Because
the direction changes, there is a change in velocity.) When a force is
applied at some other angle than a right angle, both the object’s speed and
the direction of its motion changes. 3.
Two-Dimensional
Motion with Gravity Idea: When an object is moving horizontally
at a constant velocity through space and the only force acting on it is the
gravitational force, the horizontal component of its velocity remains
constant and the vertical component of its velocity increases at a constant
rate of 9.8 m/s/s. The object follows
a parabolic path.
4.
Circular
Motion and Force Idea: An object that is moving in a
circular path at constant speed has a force applied at right angles to its
motion at all times. This force is
called the centripetal force and
it always acts towards the center of the circular path. (Centripetal means
“center-seeking”.) The centripetal
force is directly proportional to the mass of the object. The direction of the
velocity of the object continually changes, but the magnitude of its velocity
(the speed) remains constant because the centripetal force is acting at right
angles to the direction of motion of the object. The direction of the object’s velocity is always at right
angles to the radius of the circle, along the tangent to the circle at the
point where the object is located.
5.
Circular
Motion and Acceleration Idea: The direction of
the acceleration of an object that is moving in a circular path is always in
the same direction as the centripetal force, towards the center of the circle. (Newton’s Second Law of motion, F = ma, also works for circular
motion.) This acceleration is called
the centripetal acceleration. If the speed of the
object increases, the centripetal acceleration increases. If the radius of the
object increases, the centripetal acceleration decreases. (Actually, the
magnitude of the acceleration is directly proportional to the square of the
velocity and inversely proportional to the radius. |
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x, pronounced “delta x,” is:
