1. Newton’s second law
In this lesson, students learn to apply Newton's second law to calculate forces from motion, and motion from forces. The lesson includes an investigation using interactive graphs. The goal of the interactive investigation is to strengthen understanding of both force and acceleration in terms of their effects on motion graphs.

2. Objectives
Explain the effect of a net force on motion using the concept of acceleration.
Calculate acceleration in units of m/s2 when given mass and force.
Use the second law to control motion models and graphs to meet predetermined goals.
The lesson objectives state what a student should know or be able to do.

3. Assessment
A net force of 10 N acts on a cart on a straight track. Label each statement below as true, possibly true, or false.
The cart moves with constant velocity.
The cart moves with constant acceleration.
The cart speeds up.
The cart slows down.
This first assessment is keyed to the first objective: Explain the action of a net force on motion using the concept of acceleration.
It will be repeated at the end of the lesson, followed by the answer.

4. Assessment
A 10 kg object is subject to a net force of 25 N. What is the acceleration of the object in m/s2?
If the object starts at rest, then how long will it be before its velocity is 25 m/s?
What is the minimum force required to increase the speed of a 1,000 kg vehicle by 10 m/s in 3 seconds?
This second assessment is keyed to the second objective: Calculate acceleration in units of m/s2 when given mass and force.
It will be repeated at the end of the lesson, followed by the answer. (The third objective is addressed in the investigation.)

5. Physics terms acceleration force Newton’s second law
New or key vocabulary appears on this slide.

6. Equations
The new equation—Newton’s second law—is shown in two forms.

7. Newton’s second law Net force (N) Acceleration(m /s2) Mass (kg)
The acceleration of an object equals the net force divided by the mass.
The second law is introduced. The mathematics is also translated into an English sentence.

8. How does the second law show that this statement is true?
Newton’s second law
Net force (N)
Acceleration(m /s2)
Mass (kg)
Velocity must change if a net force acts on an object.
How does the second law show that this statement is true?
The second law is introduced.

9. The meaning of the second law
Net force (N)
Acceleration(m /s2)
Mass (kg)
Velocity must change if a net force acts on an object.
According to the second law, a net force on an object causes it to accelerate. If an object accelerates, its velocity must change.
The second law is introduced.

10. How does the second law show that this statement is true?
The meaning of the second law
Net force (N)
Acceleration(m /s2)
Mass (kg)
The net force is zero on an object with constant velocity.
How does the second law show that this statement is true?
The second law is introduced.

11. The meaning of the second law
Net force (N)
Acceleration(m /s2)
Mass (kg)
The net force is zero on an object with constant velocity.
If velocity stays constant, then the acceleration is zero—so the net force must also be zero.
The second law is introduced.

12. Acceleration and force are vectors.
Direction of force and acceleration
Acceleration and force are vectors.
If the student knows the direction of the net force on an object, he/she knows that the object must also have an acceleration IN THAT DIRECTION.

13. Test your knowledge
A student drags a 10 kg box across a rough level floor with a constant velocity of 1.5 m/s.
What is the net force on the box?
The key here is not the 10 kg or 1.5 m/s. The key is constant velocity. Students should learn to look for these words in a problem. One of the first questions they should always ask themselves when solving dynamics problems is this: “is the object accelerating?” Another way to phrase this: “Is this a first law situation or a second law situation?”

14. Test your knowledge
A student drags a 10 kg box across a rough level floor with a constant velocity of 1.5 m/s.
What is the net force on the box?
The net force on the box is zero!

15. What if the box has a constant velocity of 10,000 m/s?
Test your knowledge
What if the box has a constant velocity of 10,000 m/s?
A student drags a 10 kg box across a rough level floor with a constant velocity of 1.5 m/s.
What is the net force on the box?
The net force on the box is zero!

16. What if the box has a constant velocity of 10,000 m/s?
Test your knowledge
What if the box has a constant velocity of 10,000 m/s?
A student drags a 10 kg box across a rough level floor with a constant velocity of 1.5 m/s.
What is the net force on the box?
The net force on the box is zero!
The net force on the box is still zero!
Point out that it doesn’t not matter how fast an object is going. If the velocity is constant, the net force is zero. if the velocity is changing, there must be a non-zero net force.

17. Test your knowledge
A student drags a 10 kg box across a rough level floor with a constant acceleration of 1.5 m/s.
Now what is the net force on the box?

18. Test your knowledge Easy! Fnet = ma = 15 N
A student drags a 10 kg box across a rough level floor with a constant acceleration of 1.5 m/s.
Now what is the net force on the box?

19. Units The second law can help you remember the definition of a newton.
Always use mass in kilograms and acceleration in m/s2 when applying Newton’s second law.

20. Exploring the ideas
Click this interactive calculator on page 141.

21. Engaging with the concepts
You can solve for force, mass, or acceleration.
Use the calculator to answer sample questions.
Click the [Run] button to see the hand apply a force to push the ball.

22. Engaging with the concepts
A net force of 500 N acts on a 100 kg cart. What is the acceleration?
If you double the mass of the cart, what is the acceleration?
500
100
What if the force is doubled instead?
Have students complete the other sample problems in the calculator.

23. Engaging with the concepts
A net force of 500 N acts on a 100 kg cart. What is the acceleration? 5 m/s2
If you double the mass of the cart, what is the acceleration? m/s2 5
500
100
What if the force is doubled instead? 10 m/s2
Point out that acceleration is directly proportional to force, and inversely proportional to mass.
Have students complete the other sample problems in the calculator.

24. Exploring the ideas
In Investigation 5A you will explore how changes to the variables in Newton’s second law affect the motion graphs.
Click this interactive simulation on page 144.

25. Investigation Part 1: Modeling the action of a force
The interactive model shows position and velocity vs. time graphs.
Red circles on the position- time graph are targets.
Adjust initial velocity v0, force F, and mass m so the curves hits both targets.

26. Investigation Part 1: Modeling the action of a force
[SIM] runs the simulation.
[Stop] stops it without changing values.
[Clear] resets all variables to zero.
[Reset] resets all variables and sets new targets.

27. Investigation Part 1: Modeling the action of a force
Enter values in the white boxes. The top score of 100 is achieved by hitting the center of each target.
Try another problem: [Reset]
How high can you get?

28. Investigation Part 1: Modeling the action of a force
Upload your solution to a real ErgoBot to observe the motion.
[Print] a copy of your solution and score.

29. Investigation Part 1: Modeling the action of a force
Sketch solution graphs with the following characteristics:
Negative v0 and positive force.
Positive v0 and negative force.
You may have to click [Reset] a few times to get targets that allow each type of solution.

30. A tougher challenge Part 2: Dynamic modeling
The second interactive simulation allows you to change the force at four different time intervals, but there are also four target circles.

31. A tougher challenge Part 2: Dynamic modeling
The advanced simulation allows you to change the force at four different time intervals, but there are also four target circles.
Can you solve a problem while keeping the velocity between -1.0 and +1.0 m/s?

32. Applying Newton’s second law
If you know the force on an object, you can predict changes in its motion.
If you know the acceleration of an object, you can determine the net force on it.
Point out that knowing the forces on an object allows you to predict its motion. AND the reverse is also true: the motion of an object can help you figure out the forces acting on it. The next set of slides will show students how to go in both directions.

33. Finding motion from forces
If you know the force on an object, you can predict changes in its motion.
A 0.25 kg ball is traveling 40 m/s to the right when it is hit with a force of 3,000 N for seconds. What is its final velocity?
If you know the force on an object, then you can predict changes in its motion.

34. Steps
A 0.25 kg ball is traveling 40 m/s to the right when it is hit with a force of 3,000 N for seconds. What is its final velocity?
Use force and mass to find acceleration through the second law.
Use the acceleration to find the change in velocity or position.

35. Solution Impacts can cause very large accelerations for short times!
A 0.25 kg ball is traveling 40 m/s to the right when it is hit with a force of 3,000 N for seconds. What is its final velocity?
Use force and mass to find acceleration through the second law.
Impacts can cause very large accelerations for short times!

36. Solution The ball reverses direction!
A 0.25 kg ball is traveling 40 m/s to the right when it is hit with a force of 3,000 N for seconds. What is its final velocity?
Use force and mass to find acceleration through the second law.
Use the acceleration to find the change in velocity or position.
Point out to the students that acceleration is the key variable that lets them link the force on the ball to its velocity.
The ball reverses direction!

37. Applying Newton’s second law
If you know the force on an object, you can predict changes in its motion.
If you know the acceleration of an object, you can determine the net force on it.

38. Finding forces from motion
If you know the acceleration of an object, you can determine the net force acting on it.
A 70,000 kg aircraft reaches a takeoff velocity of 67 m/s (150 mph) in 11 seconds. Calculate the minimum force required from the engines.
If you know the acceleration of an object, then you can calculate the net force on it.

39. Steps
A 70,000 kg aircraft reaches a takeoff velocity of 67 m/s (150 mph) in 11 seconds. Calculate the minimum force required from the engines.
Use velocity, distance, and time find the acceleration.
Use the acceleration and mass to find the force.

40. Solution
A 70,000 kg aircraft reaches a takeoff velocity of 67 m/s (150 mph) in 11 seconds. Calculate the minimum force required from the engines.
Use velocity, distance, and time find the acceleration.

41. Solution This is almost 2/3 of the aircraft’s weight!
A 70,000 kg aircraft reaches a takeoff velocity of 67 m/s (150 mph) in 11 seconds. Calculate the minimum force required from the engines.
Use velocity, distance, and time find the acceleration.
Use the acceleration and mass to find the force.
This is almost 2/3 of the aircraft’s weight!

42. Assessment
A net force of 10 N acts on a cart on a straight track. Label each statement below as true, possibly true, or false.
The cart moves with constant velocity.
The cart moves with constant acceleration.
The cart speeds up.
The cart slows down.
This first assessment is keyed to the first objective: Explain the action of a net force on motion using the concept of acceleration.
The answers appear on the next slide.

43. Assessment
A net force of 10 N acts on a cart on a straight track. Label each statement below as true, possibly true, or false.
The cart moves with constant velocity. false
The cart moves with constant acceleration. true
The cart speeds up possibly true
The cart slows down. possibly true

44. Assessment
A 10 kg object is subject to a net force of 25 N. What is the acceleration of the object in m/s2?
This second assessment is keyed to the second objective: Calculate acceleration in units of m/s2 when given mass and force.
The answer appears on the next slide.

45. Assessment
A 10 kg object is subject to a net force of 25 N. What is the acceleration of the object in m/s2?
The second law says a = F/m. Therefore a = 25 N /10 kg = 2.5 m/s2
If the object starts at rest, then how long will it be before its velocity is 25 m/s?

46. Assessment
A 10 kg object is subject to a net force of 25 N. What is the acceleration of the object in m/s2?
The second law says a = F/m. Therefore a = 25 N /10 kg = 2.5 m/s2
If the object starts at rest, then how long will it be before its velocity is 25 m/s?
You know that v = v0 + at and v0= Rearranging gives t = v/a = (25 m/s) / (2.5 m/s2) = 10 seconds.
The answer appears on the next slide.

47. Assessment
What is the minimum force required to increase the speed of a 1,000 kg vehicle by 10 m/s in 3 seconds?

48. Assessment F = ma = (1,000 kg)( 10 m/s / 3 s) = 3,333 N
What is the minimum force required to increase the speed of a 1,000 kg vehicle by 10 m/s in 3 seconds?
F = ma = (1,000 kg)( 10 m/s / 3 s) = 3,333 N