# PS-5.1 Explain the relationship among distance, time, direction, and the velocity of an object.

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1 PS-5.1 Explain the relationship among distance, time, direction, and the velocity of an object. It is essential for students to Understand Distance and Displacement: Distance is a measure of how far an object has moved and is independent of direction. If a person travels 40m due east, turns and travels 30m due west, the distance traveled is 70m. Displacement has both magnitude (measure of the distance) and direction. It is a change of position in a particular direction. For example: 40m east is a displacement. Total or final displacement refers to both the distance and direction of an object s change in position from the starting point or origin. Displacement only depends on the starting and stopping point. Displacement does not depend on the path taken. If a person travels 40m due east, turns and travels 30m due west, the total displacement of the person is 10m east. If a person travels 40m east and then travels another 50m east the total displacement is 90m east. Understand Speed: Speed is how fast something is going. It is a measure of the distance covered per unit of time and is always measured in units of distance divided by units of time. (The term per means divided by ) Speed is a rate as it is a change (change in distance) over a certain period of time Speed is independent of direction. The speed of an object can be described two ways o Instantaneous speed is the speed at a specific instant. Initial speed and final speed are examples of instantaneous speed. A speedometer measures instantaneous speed. o Average speed is the total distance covered in a particular time period If an object is traveling at a constant speed, the instantaneous speed at each point will be equal to the average speed. If an object is traveling with varying speeds, the average speed is the total distance covered divided by the total time. Understand Velocity: Velocity refers to both the speed of an object and the direction of its motion. A velocity value should have both speed units and direction units, such as: m/sec north, km/h south, cm/s left, or km/min down. Velocity is a rate because it is a change in displacement over a certain period of time. The velocity of an object can be changed in two ways: The speed of the object can change (it can slow down or speed up). The direction of an object can change. (A racecar on a circular track moving at a constant speed of 100 km/h has a constantly changing velocity because of a changing direction of travel.) The velocity of an object can be described two ways: Instantaneous velocity is the velocity at a specific instant. Initial velocity and final velocity are examples of instantaneous velocity. Average velocity is the total (final) displacement in a particular time. 1 / 26

2 PS-5.2 Use the formula v = d/t to solve problems related to average speed or velocity. It is essential for students to Understand the correct context for the variables in the word problem when using the equation v = d/t. o In the equation, v can represent either velocity or speed and d can represent either displacement or distance, depending on the context of the problem. The differences are addressed in PS-5.1 o The term speed or velocity refers to average speed or velocity. o Students must determine the given information in a problem using the correct units. See sample table: Variable Abbre- Units Direction Examples viation required? Speed v distance/time No direction m/s 22 cm/yr Velocity v distance/time With direction m/s north, 36 km/h west Distance d distance No direction 15m 30.0 km Displacement d distance With direction 546 km down 24.9 m west t time NA 15 s 32 days Use the formula, v = d/t. o Students must be able to calculate average speed. When calculating average speed using v = d/t: the average speed for the trip equals the total distance divided by the total time. Ignore the direction of the motion. o Students must be able to calculate average velocity. When calculating average velocity using v = d/t: the average velocity equals the total displacement divided by the total time. The total displacement may be different from the total distance. When indicating the average velocity, direction must be given and the average velocity will have the same direction as the total displacement. The total displacement is the (straight line or shortest) distance and direction from the starting point. If the direction of the motion is changing, the velocity will not be constant even if the speed is constant. o Students must be able to rearrange the equation to solve for any of the variables. Example: d = vt, or t = d/v o The instantaneous velocity at any point will not necessarily be the same as the average velocity. 2 / 26

3 PS-5.3 Explain how changes in velocity and time affect the acceleration of an object. It is essential for students to understand Constant Velocity or Zero Acceleration: The first motion diagram shown below is for an object moving at a constant speed toward the right. The motion diagram might represent the changing position of a car moving at constant speed along a straight highway. Each dot indicates the position of the object at a different time. The dots are separated by equal time intervals. Because the object moves at a constant speed, the displacements from one dot to the next are of equal length. The velocity of the object at each position is represented by an arrow. The velocity arrows are of equal length (the velocity is constant). The acceleration in the diagram below is zero because the velocity does not change. Below is a data table which shows an example of what instantaneous velocities might be if measured at equal time intervals for zero acceleration. Notice the velocity is the same each time. Initial time After one second After two seconds After three seconds After four seconds Instantaneous velocity 15 m/s to the right 15 m/s to the right 15m/s to the right 15m/s to the right 15m/s to the right Constant Positive Acceleration (speeding up): This motion diagram represents an object that undergoes constant acceleration toward the right in the same direction as the initial velocity. This occurs when the car speeds up to pass another car. Once again the dots represent, schematically, the position of the object at equal time intervals. Because the object accelerates toward the right, its velocity arrows increase in length toward the right as time passes. The distance between adjacent positions increases as the object moves right because the object moves faster as it travels right. The acceleration in the diagram below is positive because the object is speeding up. Below is a data table which shows an example of what instantaneous velocities might be if measured at equal time intervals for positive acceleration. Notice the velocity is greater each time. Initial time After one second After two seconds After three seconds After four seconds Instantaneous Velocity 0 m/s to the right 5 m/s to the right 10 m/s to the right 15 m/s to the right 20 m/s to the right 3 / 26

4 Constant Negative Acceleration (slowing down): This type of motion occurs when a car slows down. The dots represent schematically the position of the object at equal time intervals. Because the acceleration is opposite the motion, the object's velocity arrows decrease by the same amount from one position to the next. Because the object moves slower as it travels, it covers less distance during each consecutive time interval, so the distance between adjacent positions decreases as the object moves right. The acceleration in the diagram below is negative because the object is slowing down. Below is a data table which shows an example of what instantaneous velocities might be if measured at equal time intervals for negative acceleration. Notice the velocity is smaller each time. Initial time After one second After two seconds After three seconds After four seconds Acceleration due to a change in direction: Initial time After one second After two seconds After three seconds After four seconds Instantaneous Velocity 20 m/s to the right 15 m/s to the right 10 m/s to the right 5 m/s to the right 0 m/s to the right Instantaneous Velocity 0 m/s 5.0 m/s north 5.0 m/s west 5.0 m/s south 5.0 m/s east Students should understand that the velocity of the object above is changing because the direction is changing. The speed of the object remains constant. Because the velocity of the object is changing, it is accelerating; Students need only say that the object is accelerating because the direction (and therefore the velocity) of the object is changing. Students need not consider the rate of acceleration for an object that is changing direction. It is essential for the students to understand That acceleration is a measure of the change in velocity (final velocity - initial velocity) per unit of time. When the velocity of an object is changing, it is accelerating. That if the object slows down, the change in velocity (v f - v i ) is negative so the acceleration is negative and conversely when the object is speeding up the acceleration is positive. That both the change in velocity and the time it takes for that change to occur are important when considering the acceleration of an object. o When comparing the acceleration of two objects that have the same change in velocity, the one that undergoes the change in the least amount of time has the greatest acceleration. 4 / 26

5 o When comparing the acceleration of two objects that accelerate over the same interval of time, the one that undergoes the greatest change in velocity accelerates the most. That acceleration is always measured in velocity (distance/time) units divided by time units. Example: Acceleration is change in velocity divided by time. The unit for velocity is m/s and the unit for time is second so the unit for acceleration is m/s/s or m/s 2. This is derived from velocity (m/s) divided by time (s). Students should understand acceleration units conceptually as change in velocity over time rather than distance over time squared. o The most common acceleration units in the metric system are m/s/s or m/s 2. o The time units may be different in the velocity part of the equation and denominator such as km/hr per second. The velocity of an object can change two ways, so an object can accelerate in two ways: The speed can increase or decrease The direction can change. It is not essential for students to solve mathematical problems involving acceleration for an object that is changing direction. PS-5.4 Use the formula a = (v f -v i )/t to determine the acceleration of an object. It is essential for students to Interpret a word problem, or laboratory data, involving the motion of an object that is accelerating in one direction and determine the given information: Differentiate velocity from speed if the direction is given. If velocity is given, students should record the direction. Differentiate initial velocity (speed) from final velocity (speed) from the context of the problem. Students need to list the given variables using the correct units: Variable Symbol Examples of units for velocity (or speed) Initial velocity (or speed) v i = distance/time 5.0 m/s east (5.0 m/s) 5.0 km/h east (5.0 km/h) Final velocity (or speed) v f = distance/time 2.0 m/s east (2.0 m/s) 2.0 km/s east (2.0 m/s) Elapsed time t 15 s 15 s It is essential for students to Use the equation a = (v f - v i )/t to solve for acceleration only, not for v f or v i. Substitute the correct values into the equation, including the correct units. Mathematically solve the problem, using dimensional analysis to derive the units of the answer. (see dimensional analysis PS-1.5) Check to make sure that the units calculated from the dimensional analysis match the appropriate units for the acceleration (distance/time divided by time or distance divided by time-squared). Understand that negative acceleration means that velocity is decreasing. 5 / 26

6 PS-5.5 Explain how acceleration due to gravity affects the velocity of an object as it falls. It is essential for students to understand that All objects accelerate as they fall because Earth continually exerts a force (gravitational force) on them. The diagram depicts the position of an object freefall at regular time intervals. The fact that the distance which the ball travels every interval of time is increasing is a sure sign that the ball is speeding up as it falls downward. If an object travels downward and speeds up, then it accelerates downward. When an object is released it accelerates. The direction of the gravitational force is always downward. The acceleration is in the direction of the force, so the direction of the acceleration is downward as well. When an object is dropped from rest, it has an initial velocity of 0.0 m/s. The object will accelerate at a constant rate of 9.8m/s 2 or m/s/s. This means that the object will speed up at a constant rate of 9.8 m/sec every second it is falling in the absence of air resistance. The value, 9.8m/s per s, is called the acceleration of gravity and has the symbol a g. Since the object is accelerating because of the gravitational force that is attracting Earth and the object, the velocity of the object continues to increase in speed and continues to fall in a downward direction until it hits the ground. Students must understand the meaning of the values on the chart in terms changing velocity. v i v f 1 st s 0.0 m/s 9.8 m/s 2 nd s 9.8 m/s 19.6 m/s 3 rd s 19.6 m/s 29.4 m/s 4 th s 29.4 m/s 39.2 m/s 5 th s 39.2 m/s 49.0 m/s Students may use 10 m/s/s for acceleration due to gravity for ease of calculation. Both 10 or 9.8 may be used on the end-of-course examination. 6 / 26

7 distance (meters) distance (meters) Forces and Motion PS-5.6 Represent the linear motion of objects on distance-time graphs. It is essential for students to Elapsed (s) Example: Elapsed (s) Total Distance Traveled (meters) (1) An object at rest Total Distance Traveled (meters) Construct distance/time graphs from data showing the distance traveled over time for selected types of motion (rest, constant velocity, acceleration). Compare the shape of these three types of graphs and recognize the type of motion from the shape of the graph. Discuss in words the significance of the shapes of the graphs in terms of the motion of the objects Object at Rest time (seconds) The shape of the graph is flat, because between the 1 st and 6 th second there is no change in distance. 2) An object with constant speed Example: Constant Speed time (seconds) The shape of the graph is a diagonal straight line. The object covers the same amount of distance in each time period. As the time increases, the distance increases at a constant rate. 7 / 26

8 distance (meters) Forces and Motion (3a) An accelerating object (positive acceleration or speeding up) Example: Accelerating Object time (seconds) The shape of the graph is a curve getting steeper because as time goes by, the object covers more distance each second than it did in the previous second so the amount that the graph goes up each second gets more and more. Elapsed (s) Total Distance Traveled (meters) / 26

9 distance (meters) distance (meters) Forces and Motion (3b) A negatively accelerating object (an object slowing down) Example: Elapsed (s) Total Distance Traveled (meters) Negative acceleration time (seconds) The shape of the graph is a curve getting flatter because as time goes by, the object covers less distance each second than it did in the previous second, so the amount that the graph goes up each second gets less and less. It is essential for students to Construct distance time graphs from data that compare the motion of objects. Discuss the significance of the shapes of the graphs in terms of the relative motion of the objects. (1) A comparison of two objects traveling at different speeds Example: Elapsed Total Distance Traveled (meters) (s) Object 1 Total Distance Traveled (meters) Object Comparison of Speed time (seconds) Both objects are traveling at a constant speed, but the object represented by the top line is traveling faster than the lower one. You can tell this because the amount that the graph goes up each second (which represents the amount of distance traveled) is more for the top line than for the bottom one. (The top line has a greater slope.) 9 / 26

10 distance (meters) Forces and Motion (2) A comparison of two objects accelerating at different rates Example: Total distance Total distance Total elapsed traveled (meters) traveled (meters) (seconds) Object 1 Object s 5.00 m m 2.00 s m m 3.00 s m m 4.00 s m m 5.00 s m m 6.00 s m m Acceleration Comparison time (seconds) Series1 Series2 Both of the objects are accelerating, but the Series 2 object (top curve) is accelerating at a greater rate than the Series 1 object (bottom curve). Both objects cover more distance each second than they did during the previous second, but the amount of increase for series 2 is more than the amount of increase for (series 1). 10 / 26

11 Displacement (m) Displacement (km) Displacement (km) Forces and Motion (3) A comparison of two objects traveling in different directions at a constant speed (to show this, a displacement-time graph is required) Example: Elapsed (s) Total Displacement (kilometers) Object 1 Total Displacement (kilometers) Object km West km West km West km West km West km West km West km West km West km West Direction Comparison Object 1 Object (hours) (hours) These are displacement-time graphs (displacement/location has distance and direction), so it shows how far each object is from the starting point after each hour. Object 1 gets farther and farther away. At the 3 rd hour, object 2 turns around and comes back toward the start. The speed of each object is the same. It is essential for a student to infer a possible story given a graph similar to this example. Story Graph (s) 11 / 26

12 Possible explanation. From 0 to 3 seconds the object is traveling at a constant velocity away from the starting point. From 3 seconds to 5 seconds the object is not moving relative to the starting point. From 5 seconds to 8 seconds the object is moving at a constant velocity toward the starting point. From 8 seconds to 13 seconds the object is moving at a constant velocity away from the starting point, at a velocity slower than the motion from 0 to 3 seconds. From 13 to 15 seconds the object is not moving relative to the starting point. From 15 to 21 seconds the object is accelerating (speeding up) as it moves away from the starting point. PS-5.7 Explain the motion of objects on the basis of Newton s three laws of motion: inertia; the relationship among force, mass, and acceleration; and action and reaction forces. Newton s First Law of Motion It is essential for students to understand That a force is a push or a pull that one object exerts on another object and that in the metric system, force is measured in units called newtons (N). That a net force is an unbalanced force. It is necessary to find the net force when one object has more than one force exerted on it. Newton s First Law that states that the velocity of an object will remain constant unless a net force acts on it. This law is often called the Law of Inertia. o If an object is moving, it will continue moving with a constant velocity (in a straight line and with a constant speed) unless a net force acts on it. o If an object is at rest, it will stay at rest unless a net force acts on it. o Inertia is the tendency of the motion of an object to remain constant in terms of both speed and direction. That the amount of inertia that an object has is dependent on the object s mass. The more mass an object has the more inertia it has. That if an object has a large amount of inertia (due to a large mass): o It will be hard to slow it down or speed it up if it is moving. o It will be hard to make it start moving if it is at rest. o It will be hard to make it change direction. That inertia does not depend on gravitational force. Objects would still have inertia even if there were no gravitational force acting on them. The behavior of stationary objects in terms of the effect of inertia. Examples might include: o A ball which is sitting still will not start moving unless a force acts on it. o A ball with a larger mass will be more difficult to move from rest than a smaller one. It is more difficult to roll a bowling ball than a golf ball. 12 / 26

13 The behavior of moving objects in terms of the effect of inertia. Examples might include: o People involved in a car stopping suddenly: If a net force (braking force) is exerted on the car in a direction opposite to the motion, the car will slow down or stop. If the people in the car are not wearing their set belts, because of their inertia, they keep going forward until something exerts an opposite force on them. The people will continue to move until the windshield (or other object) exerts a force on them. If the people have their seatbelts on when the braking occurs, the seatbelt can exert a force to stop the forward motion of the person. A passenger in a turning car: Consider a person who is a passenger in a car that is moving in a straight path. The passenger in the car is also moving in a straight path. If the car suddenly turns left, the inertia of the passenger causes him to continue to move in the same straight path even though the car under him has turned to the left. The passenger feels as if he has been thrown against the side of the car, but in fact, the car has been pushed against the passenger. If a rowboat and a cruise ship are moving at the same speed, it is more difficult to turn the cruise ship because it has more mass and therefore more inertia. The reason that objects often do not keep moving in our everyday experience is because there is often a net force acting on them. Students need to explain how friction as a net force slows or stops a variety of everyday objects. If a ball were thrown in distant outer space away from forces, such as friction, it would continue to move at a constant velocity until an outside force acts on it. Newton s Second Law of Motion It is essential for students to understand Newton s Second Law that states, When a net force acts on an object the object will accelerate in the direction of the net force. The larger the net force, the greater the acceleration. (It is sometimes stated that the acceleration is directly proportional to the net force.) The larger the mass of the object, the smaller the acceleration. (It is sometimes stated that the acceleration is inversely proportional to the mass of the object.) In mathematical terms Newton s Second Law states that the net force equals the mass times the resulting acceleration. (F = ma) Acceleration can mean speeding up, slowing down, or changing direction; Friction and air resistance will often be ignored in discussions and problems, but students should be aware of their role in determining the net force. It is essential that students understand The motion of objects in terms of force, mass and acceleration. The effects of force: o Force magnitude: If the mass of an object remains constant, the greater the net force the greater the rate of acceleration. 13 / 26

14 o Force direction: If the force is applied to an object at rest, the object will accelerate in the direction of the force. If the force is applied to a moving object in the same direction that the object is moving, the object will accelerate so its speed will increase to a greater speed and continue to travel in the same direction. If the force is applied to a moving object in a direction opposite to the direction that the object is moving, the object will have negative acceleration and slow down from its speed before the force was applied to a slower speed. It will either continue at the slower speed, stop, or begin to move in the opposite direction, depending on the magnitude of the force. The effect of mass: o If the same net force is applied to two different objects, the object with the smaller mass will have a greater acceleration in the direction of the applied force. Newton s Third Law of Motion It is essential for students to understand Newton s Third Law that states, When one object exerts a force on a second object, the second one exerts a force on the first that is equal in magnitude and opposite in direction. This law is sometimes called the Law of Action and Reaction. Even though the forces are equal in magnitude and opposite in direction, they do not cancel each other. This law addresses two objects, each with only one force exerted on it. Each object is exerting one force on the other object. Each object is experiencing only one force. Object 1 Object 2 Students should describe the motion of familiar objects in terms of action and reaction forces. Examples may include: A swimmer is accelerating forward: The swimmer pushes against the water (action force), the water pushes back on the swimmer (reaction force) and pushes her forward. A ball is thrown against a wall: The ball puts a force on the wall (action force), and the wall puts a force on the ball (reaction force) so the ball bounces off. A person is diving off a raft: The person puts a force on the raft (action force) pushing it, and the raft puts a force on the diver (reaction force) pushing her in the opposite direction. A person pushes against a wall (action force), and the wall exerts an equal and opposite force against the person (reaction force). It is not essential for students to understand or solve problems involving momentum. 14 / 26

15 PS-5.8 Use the formula F = ma to solve problems related to force. It is essential for students to Understand the correct context for the variables in a word problem. Understand that a newton is defined as the amount of force necessary to accelerate a 1.0 kg object at a rate of 1 meter/second/second. force = (mass)(acceleration) o The newton is a derived unit, so when you multiply mass times acceleration, if mass is in kilograms and acceleration is in m/s/s, you have the proper units for newtons (kgm/s/s or kgm/s 2 ). Mathematically solve problems for force, mass, or acceleration, using dimensional analysis to identify the units of the answer. (See dimensional analysis PS-1.5) Determine the given information using the correct units, o Mass should be given in kilograms (kg), o Acceleration in (m/s/s, or m/s 2), and o Force in newtons. (N) Solve problems for any of the variable in the formula, F = ma. For example, the problem may give net force and mass and the student must find the acceleration (a = F/m). PS-5.9 Explain the relationship between mass and weight by using the formula F W = ma g. It is essential for students to understand The weight of an object is the force that gravity exerts on that object. o The weight of an object depends on its mass. o Given the mass of an object, its weight can be calculated using Newton s Second Law. o When an object is dropped, it accelerates at 9.8m/s 2. Because there is acceleration, there must be a force. o The force is equal to the mass times the acceleration. (F w =ma g ) o The force called weight is equal to an object s mass times the acceleration due to gravity. (9.8m/s 2 ) It is essential for students to Solve problems involving the relationship among the weight and mass of objects and the acceleration of gravity using the formula F w =ma g. (This formula is sometimes written, w = mg.) PS-5.10 Explain how the gravitational force between two objects is affected by the mass of each object and the distance between them. It is essential for students to understand Newton s Law of Universal Gravitation states that there is a force of attraction between all objects in the universe. The Law of Universal Gravitation applies to all objects. The size of the gravitational force of attraction between two objects depends on the mass of both objects and the distance between objects. The force is greater when the mass of either of the two objects is greater. Earth, with its huge mass, has a relatively large attractive force with all of the objects near its surface. The moon has less mass than Earth, so the moon has less attraction for objects on its surface than Earth does. (Objects on the surface of the moon weigh less than on Earth because the gravitational 15 / 26

16 force between the object and the moon is less than the gravitational force between the object and the Earth.) The reason the attraction is not noticed between ordinary sized objects that are on earth is that the force that Earth exerts on objects is so great relative to the force of attraction between other objects. (negligible relative to the attraction of the object to Earth). The closer the two objects are, the greater the force When an object, such as a space vehicle, moves away from Earth, the gravitational attraction between Earth and the vehicle becomes less and less. That if the force acting on a falling object is the same as the force acting on Earth, the object accelerates toward Earth while Earth doesn t seem to accelerate at all. This is because the mass of Earth is so huge, the force causes only a very tiny acceleration, one that is undetectable by humans. 16 / 26

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