Course: 8^th grade Physics

Unit Introduction: Force, Motion, and Energy

The Motion, Force and Energy Unit engages students with many hands-on and computer-based experiences that will help them modify their existing ideas and construct new ideas about the motion, forces, and their relation to each other. This unit consists of six parts. The first two parts use computer aided data collection and simulators to help students represent motion; Part 1 focuses on motion in one dimension and one direction, Part 2 focuses on motion in one dimension and two directions. In Part 3 students explore the relationships between motion and forces. In Part 4 students use bathroom scales, spring scales, simulators, and the data collectors to get a sense of equal and opposite forces and conservation of momentum. Part 5 focuses on gravity, balanced forces and inertia in the vertical direction. The focus of Part 6 is on energy transfer and energy conservation.

Objectives: The students learn that…

Part 1:

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 curving upward. If the object is slowing down, the curve is curving downward. Objects that are not moving have a distance-time graph that is a horizontal straight line.

· Given a tangent line for a curved distance-time graph, the instantaneous speed at that time can be calculated from the slope of the tangent line.

· The average speed is found by calculating the slope of the line segment joining the two points on the graph that correspond to the time interval.

1. 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.

· For an object moving at a constant speed, 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. 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 at a constant rate 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 changing speed at a constant rate can be found by taking the slope of a speed-time graph.

Part 2:

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 away from the motion detector (to the right) and negative when the object is moving towards the motion detector (to the left).

Students should be able to draw and interpret the following graphs using the ideas above:

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. Students should be able to draw and interpret the following graphs using the ideas above:

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 x, pronounced “delta x,” is:

· 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:

· Given a tangent line, the value of the instantaneous velocity at an instant of time can be calculated from 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 velocity axis 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 can be calculated from the slope of the velocity-time graph between two points. In symbols:

· Given positive positions for the object, positive acceleration either means speeding up in the original direction (away from the reference point), or slowing down in the opposite direction (towards the reference point).

· Given positive positions for the object, negative acceleration either means slowing down in the original direction (away from the reference point), or speeding up in the opposite direction (towards the reference point).

Students should be able to draw and interpret the following graphs using the ideas above:

Part 3:

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, and the push between the two surfaces. Generally, the rougher the surfaces the stronger the friction force, the smoother the surfaces, the weaker the friction force. The stronger the push between the two surfaces, the stronger the friction force.

· The stronger the friction force, the more quickly an object will slow down.

2. Object at Rest Idea: If the forces acting on an object at rest are balanced, the object will remain at rest.

3. Combining Forces Idea: When two or more forces act on the same object, they combine such that forces in the same direction add to each other, forces in the opposite direction subtract from each other.

· If the forces are in opposite directions and are the same strength, the combined forces are said to be balanced.

· If they are not the same strength, the combined forces are said to be unbalanced.

4. Motion with an Unbalanced Force Idea: If the forces acting on an object are unbalanced, its speed will change continuously causing it to accelerate in the same direction as the larger force.

· If the force is constant, the acceleration will be constant.

· The acceleration is directly proportional to the force and inversely proportional to the mass. This can be written in the form of a mathematical equation.

Either or

This is also known as Newton’s Second Law of Motion

5. Mass Idea: The more massive an object, the harder it is to get it moving when it is at rest (“holding back” property), and the harder it is to stop it when it is moving (“keep going” property).

· The inertia of an object is the tendency for an object to resist a change in its state of motion.

· If it is at rest, it tends to stay at rest.

· If it is in motion, it tends to stay in motion at a constant speed, the more mass an object has, the more inertia it has, and the more difficult it is to change its motion.

6. Motion without an Unbalanced Force Idea: No unbalanced force means the forces are balanced.

· If the object is at rest and there are no unbalanced forces acting on the object, the object will stay at rest.

· If the object is moving and there are no unbalanced forces acting on the object, the object will continue moving at constant speed.

This is Newton’s first law of motion. When the forces on an object are balanced, the object will either remain at rest or move at a constant speed.

Part 4:

1. Comparative Strengths of Forces of Interacting Objects Idea: All forces involve interactions between two 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 in direction. Each force acts on the only one of the interacting objects.

· 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 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 during a time interval, 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.

(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: . 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.

5. Total Momentum During Interactions Idea: When two objects interact, the sum of the individual momentums of the two objects before the interaction is equal to the sum of the individual momentums 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), or whether they bounce off each other (elastic collision).

Part 5:

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 appear 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 strength of the electric force can be increased by rubbing harder or longer.

· 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 toward the center of the Earth (the observer experiences this as downward) and its strength is proportional to the amount of mass contained in the object.

Acceleration Due To Gravity Idea: If the gravitational force between the Earth and an object is the only (significant) force acting on an object, then all objects, regardless of mass, will fall with the same acceleration of 9.80 meters per sec per sec. This means that each second its velocity increases by 9.80 meters per sec (until it hits the ground).
Vertical Inertia Idea: The larger the mass of an object, the more difficult it is to change its motion in the vertical direction (holding-back and keep-going property of objects).

· 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 è Big Inertia è More Force Needed

Small mass è Small Inertia è Less Force Needed

or for a falling object.

· To have the same acceleration, if m is bigger, F must be bigger, if m is smaller, F must be smaller.

Weight Idea: The weight of an object is measured using a scale, in which the supporting force provided by the scale on the object (the scale reading), exactly balances the gravitational force on the object, when there are only two significant forces acting on the object.

· Mass is different from weight. Mass is the amount of stuff contained in the object, and is a property of the object. Weight is a measure of the gravitational force of the Earth on the object.

· Mass has units of kilograms and weight has units of Newtons.

· The weight of an object is equal to the gravitational force acting on it.

Balanced Vertical Forces Idea: If an object is to remain at rest, then all forces in the vertical direction, including the gravitational force, must balance each other out. (In addition, all forces in the horizontal direction must balance each other out.)

· 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

Students should be able to interpret the information above from diagrams such as these:.

Part 6:

1. Force and Distance Idea

· The greater the braking force, the less distance required to stop the object. The braking force applied to an object and the stopping distance are inversely proportional,

The simulator snapshot below shows data giving the stopping distance for four different braking forces. The mass of the car (6 kg) and the initial velocity of the car (40 m/s) were kept constant for the four trials. When the braking force is doubled, the stopping distance is 1/2 as large, when the braking force is tripled; the stopping distance is 1/3 as large, etc.

Work and Change in Energy Idea:

· Work can be defined as the product of force and distance. (where the distance traveled is measured in the same direction as the force.) The unit of work is the Joule.

· Kinetic energy is the energy an object possesses due to its motion.

Kinetic Energy has the same units as work, i.e., the Joule.

An object at rest has zero velocity, therefore its KE = 0

· The work done on an object is equal to the object’s change in kinetic energy.

(1) If the force (braking force or frictional force) is in the opposite direction of motion, the object slows down. If the force is in the same direction of motion, the object speeds up.

(2) When the mass and braking force are kept constant, the stopping distance is proportional to the square of the velocity.

(3) When the mass and stopping distance are kept constant, the braking force is proportional to the square of the velocity.

(4) The greater the mass of a moving object, the more inertia it possesses. Therefore a greater force is required to change its state of motion. So it is necessary to do more work on the object in order to change its state of motion, (stop it, slow it down, or speed it up). The work required to stop an object is directly proportional to the mass of the object.

(5) When the mass of the object is kept constant, the braking work done on the car is proportional to the square of the velocity,

(6) When the mass and initial velocity of the car remains constant, the work required to stop the car for different braking forces remains constant.

3. Kinetic Energy and Gravity Idea

· When an object moves under the influence of the gravitational force, the maximum height reached by the object is directly proportional to the object’s kinetic energy.

· Gravitational potential energy is defined as

where y is the height above an arbitrary reference level, usually ground level.

· If no work is done on a system, the changes in kinetic energy and gravitational potential energy will be equal in magnitude but have opposite signs, i.e., when one increases the other decreases by the same amount, so their sum is always equal to zero and no work is done. The change in KE is equal and opposite to the change in PE_g.

The following snapshots show the energy bar graphs for an object falling from rest. In each case, the change in KE is equal and opposite to the change in PE_g.

The object has fallen about 1/3 of the distance to the ground.

The object is now at ground level.

· If another force does work on an object, the total energy in the system is constant and equal to the work done on the system. At any point in time, the work done on the system is equal to the sum of the change in KE and the change in PE_g.

The four simulator snapshots below show the superimposed KE and PE_g graphs and the energy bar graphs for an object that is launched upward by the force of the launcher. The launcher does work on the object to give it an initial vertical velocity. At any point in time, as the object moves upward and then downward again, the sum of the changes in KE and PE_g is equal to the work done on the system.

Using different launcher strengths and/or different masses, still confirms that the

Friction does negative work on a moving object that causes the object to lose KE because the frictional force is opposite the direction of motion. The work done by friction equals the sum of the energy changes in the system. The following simulator snapshots show that the negative work done on the system is always equal to the sum of the changes in KE and PE_g.

The object begins to slide down the left track.

The object has reached the center track.

The object is part way up the right track.

The object has lost all of its KE and has stopped on the right track.

4. Kinetic energy and Spring Idea: When an object moves under the influence of a spring (elastic) force, it has maximum speed at the equilibrium position and minimum speed (zero speed) when it is farthest from the equilibrium position.

· At the equilibrium position the object’s elastic potential energy is equal to zero because the spring is neither stretched nor compressed. When the object is farthest from the equilibrium position (where the object’s speed is zero), the spring is stretched (or compressed) the maximum amount and therefore has the greatest elastic potential energy

· A spring (elastic) force is not constant. The spring force is directly proportional to the displacement, i.e., it takes more and more force to stretch (or compress) a spring farther and farther.

This can be written in the form of an equation if a proportionality constant is added.

where k = spring constant

The elastic potential energy is defined as

· As an object oscillates back and forth under the influence of a spring force, the sum of the KE and PE_elastic at any given point in time is constant. The simulator graph at the right shows overlapping KE and PE_elastic graphs. The sum of the KE and PE_elastic at any point in time is equal to 2.25 kJ.

· As an object gains KE, it loses an equal amount of PE_elastic and vice versa. The simulator snapshot below shows that the sum of the change in KE and the change in PE_elastic is constant and equal to zero, i.e.,

Resources:

Graphs of an object speeding up to the left = negative acceleration

Graphs of an object speeding up to the right = positive acceleration.

Graphs of an object slowing down to the right = positive acceleration

Graphs of an object slowing down to the left = negative acceleration.

Graphs of an object slowing down to the right = positive acceleration

Graphs of an object slowing down to the left = negative acceleration.

The graphs of an object that begins with an initial velocity, slows down at a constant rate to the right, stops, reverses direction, and accelerates to the left at a constant rate.