The Impulse-Momentum Change Theorem
MOMENTUM LESSON 1
MOMENTUM
The amount of momentum which an object has is dependent upon two variables: how much stuff is moving and how fast the stuff is moving.
Momentum = mass • velocity
p = m • v
The unit for momentum is kg•m/s.
From the definition of momentum, it becomes obvious that an object has a large momentum if either its mass or its velocity is large.
Both variables are of equal importance in determining the momentum of an object.
Momentum is a vector quantity. The direction of the momentum vector is the same as the direction of the velocity of the ball.
EXAMPLE: If the bowling ball is moving westward, then its momentum can be fully described by saying?
That it is ___ kg•m/s, westward.
EXAMPLE: Consider a Mack truck and a roller skate moving down the street at the same speed.
Which has greater momentum and why?
Answer: The considerably greater mass of the Mack truck gives it a considerably greater momentum.
What if the Mack truck were at rest, but the much lighter roller skate was still moving, then which would have the greater momentum?
Answer: The momentum of the less massive roller skate would be the greatest. The momentum of any object which is at rest is 0. Objects at rest do not have momentum - they do not have any "mass in motion."
Example: What happens to the momentum when you quadruple the mass?Answer: momentum quadruples
How about when you double the velocity? Answer: Momentum doubles.
Momentum and Impulse Connection
Any object with momentum is going to be hard to stop.
To stop such an object, it is necessary to apply a force against its motion for a given period of time.
The more momentum which an object has, the harder that it is to stop. Thus, it would require a greater amount of force or a longer amount of time or both to bring such an object to a halt.
As the force acts upon the object for a given amount of time, the object's velocity is changed; and hence, the object's momentum is changed.
EXAMPLE: What happens when you brake at a stop sign when your driving? Answer: the brakes serve to apply a force to the car for a given amount of time to change the car's velocity and therefore change its momentum.
What happens when you hit the gas when your driving? You are accelerating the car by applying a force to the car over a period of time. This too changes the velocity of the car and therefore changes its momentum (increases)
Newton's second law (Fnet = m • a) stated that the acceleration of an object is directly proportional to the net force acting upon the object and inversely proportional to the mass of the object. When combined with the definition of acceleration (a = change in velocity / time), the following equalities result.
[pic]
If both sides of the above equation are multiplied by the quantity t, a new equation results.
[pic]
In physics, the quantity Force • time is known as impulse.
And since the quantity m•v is the momentum, the quantity m•[pic]v must be the change in momentum.
The equation really says that the
Impulse = Change in momentum
FIRST LAW OF MOMENTUM:
If an object experiences a force for a specific amount of time there is a change in momentum. The result of the force acting for the given amount of time is that the object's mass either speeds up or slows down (or changes direction). The impulse experienced by the object equals the change in momentum of the object. In equation form, F • t = m •[pic] v.
EXAMPLE: Consider a football halfback running down the football field and encountering a collision with a defensive back. The collision would change the halfback's speed and thus his momentum.
If the motion was represented by a ticker tape diagram, it might appear as follows:
[pic]
At approximately the tenth dot on the diagram, the collision occurs and lasts for a certain amount of time; in terms of dots, the collision lasts for a time equivalent to approximately nine dots.
In the halfback-defensive back collision, the halfback experiences a force which lasts for a certain amount of time to change his momentum. Since the collision causes the rightward-moving halfback to slow down, the force on the halfback must have been directed leftward.
If the halfback experienced a force of 800 N for 0.9 seconds, then we could say that the impulse was 720 N•s.
This impulse would cause a momentum change of 720 kg•m/s. In a collision, the impulse experienced by an object is always equal to the momentum change.
EXAMPLE: Now consider a collision of a tennis ball with a wall. Depending on the physical properties of the ball and wall, the speed at which the ball rebounds from the wall upon colliding with it will vary. Which case (A or B) has the greatest change in velocity, greatest acceleration, greatest momentum change, and greatest impulse.
NOTE: Velocity is a vector quantity and has direction, if one way is +, then the opposite direction is negative.
Vector Diagram
[pic]
|Greatest velocity change? |
|Greatest acceleration? |
|Greatest momentum change? |
|Greatest Impulse? |
ANSWER: a. The velocity change is greatest in case B. The velocity changes from +30 m/s to -28 m/s. This is a change of 58 m/s (-) and is greater than in case A (-15 m/s).
b. The acceleration is greatest in case B. Acceleration depends on velocity change and the velocity change is greatest in case B (as stated above)
c. The momentum change is greatest in case B. Momentum change depends on velocity change and the velocity change is greatest in case B (as stated above).
d. The impulse is greatest in case B. Impulse equals momentum change and the momentum change is greatest in case B (as stated above).
A rebound is a special type of collision involving a direction change in addition to a speed change. The result of the direction change is a large velocity change, acceleration, momentum change, and impulse.
MOMENTUM LESSON 1 HOMEWORK
1. Determine the momentum of a ...
a. 60-kg halfback moving eastward at 9 m/s.
b. 1000-kg car moving northward at 20 m/s.
c. 40-kg freshman moving southward at 2 m/s.
2. A car possesses 20 000 units of momentum. What would be the car's new momentum if ...
a. its velocity were doubled.
b. its velocity were tripled.
c. its mass were doubled (by adding more passengers and a greater load)
d. both its velocity were doubled and its mass were doubled.
3. A halfback (m = 60 kg), a tight end (m = 90 kg), and a lineman (m = 120 kg) are running down the football field. Consider their ticker tape patterns below.
[pic]
Compare the velocities of these three players. How many times greater is the velocity of the halfback and the velocity of the tight end than the velocity of the lineman?
Which player has the greatest momentum? Explain.
4. Which case (A or B) has the greatest change in velocity, greatest acceleration, greatest momentum change, and greatest impulse.
Velocity-Time Graph
[pic]
|Greatest velocity change? |
|Greatest acceleration? |
|Greatest momentum change? |
|Greatest Impulse? |
5. Which case (A or B) has the greatest change in velocity, greatest acceleration, greatest momentum change, and greatest impulse.
Ticker Tape Diagram
[pic]
|Greatest velocity change? |
|Greatest acceleration? |
|Greatest momentum change? |
|Greatest Impulse? |
6. Use the impulse-momentum change principle to fill in the blanks in the following rows of the table. As you do, keep these three major truths in mind:
• the impulse experienced by an object is the force•time
• the momentum change of an object is the mass•velocity change
• the impulse equals the momentum change
| |Force |time |Impulse |Mom. Change |Mass |Vel. Change |
| |(N) |(s) |(N*s) |(kg*m/s) |(kg) |(m/s) |
|1. | |0.010 | | |10 |-4 |
|2. | |0.100 |-40 | |10 | |
|3. | |0.010 | |-200 |50 | |
|4. |-20 000 | | |-200 | |-8 |
7. A 0.50-kg cart (#1) is pulled with a 1.0-N force for 1 second; another 0.50 kg cart (#2) is pulled with a 2.0 N-force for 0.50 seconds. Which cart (#1 or #2) has the greatest acceleration? Explain.
Which cart (#1 or #2) has the greatest impulse? Explain.
Which cart (#1 or #2) has the greatest change in momentum? Explain.
8. In a physics demonstration, two identical balloons (A and B) are propelled across the room on horizontal guide wires. The motion diagrams (depicting the relative position of the balloons at time intervals of 0.05 seconds) for these two balloons are shown below.
[pic]
Which balloon (A or B) has the greatest acceleration? Explain.
Which balloon (A or B) has the greatest final velocity? Explain.
Which balloon (A or B) has the greatest momentum change? Explain.
Which balloon (A or B) experiences the greatest impulse? Explain.
9. Two cars of equal mass are traveling down Lake Avenue with equal velocities. They both come to a stop over different lengths of time. The ticker tape patterns for each car are shown on the diagram below.
[pic]
At what approximate location on the diagram (in terms of dots) does each car begin to experience the impulse?
Which car (A or B) experiences the greatest acceleration? Explain.
Which car (A or B) experiences the greatest change in momentum? Explain.
Which car (A or B) experiences the greatest impulse? Explain.
10. The diagram to the right depicts the before- and after-collision speeds of a car which undergoes a head-on-collision with a wall. In Case A, the car bounces off the wall. In Case B, the car crumples up and sticks to the wall.
a. In which case (A or B) is the change in velocity the greatest? Explain.
b. In which case (A or B) is the change in momentum the greatest? Explain.
c. In which case (A or B) is the impulse the greatest? Explain.
d. In which case (A or B) is the force which acts upon the car the greatest (assume contact times are the same in both cases)? Explain.
11. Jennifer, who has a mass of 50.0 kg, is riding at 35.0 m/s in her red sports car when she must suddenly slam on the brakes to avoid hitting a deer crossing the road. She strikes the air bag, which brings her body to a stop in 0.500 s. What average force does the seat belt exert on her?
If Jennifer had not been wearing her seat belt and not had an air bag, then the windshield would have stopped her head in 0.002 s. What average force would the windshield have exerted on her?
12. A hockey player applies an average force of 80.0 N to a 0.25 kg hockey puck for a time of 0.10 seconds. Determine the impulse experienced by the hockey puck.
13. If a 5.0-kg object experiences a 10.0-N force for a duration of 0.10-second, then what is the momentum change of the object?
MOMENTUM LESSON 1 HOMEWORK
1. Determine the momentum of a ...
a. 60-kg halfback moving eastward at 9 m/s.
b. 1000-kg car moving northward at 20 m/s.
c. 40-kg freshman moving southward at 2 m/s.
Answer: A. p = m*v = 60 kg*9 m/s
p = 540 kg•m/s, east
B. p = m*v = 1000 kg*20 m/s
p = 20 000 kg•m/s, north
C. p = m*v = 40 kg*2 m/s
p = 80 kg•m/s, south
2. A car possesses 20 000 units of momentum. What would be the car's new momentum if ...
a. its velocity were doubled.
b. its velocity were tripled.
c. its mass were doubled (by adding more passengers and a greater load)
d. both its velocity were doubled and its mass were doubled.
Answer: A. p = 40 000 units (doubling the velocity will double the momentum)
B. p = 60 000 units (tripling the velocity will triple the momentum)
C. p = 40 000 units (doubling the mass will double the momentum)
D. p = 80 000 units (doubling the velocity will double the momentum and doubling the mass will also double the momentum; the combined result is that the momentum is doubled twice -quadrupled)
3. A halfback (m = 60 kg), a tight end (m = 90 kg), and a lineman (m = 120 kg) are running down the football field. Consider their ticker tape patterns below.
[pic]
Compare the velocities of these three players. How many times greater is the velocity of the halfback and the velocity of the tight end than the velocity of the lineman?
Which player has the greatest momentum? Explain.
Answer: A. The tight end travels twice the distance of the lineman in the same amount of time. Thus, the tight end is twice as fast (vtight end = 6 m/s). The halfback travels three times the distance of the lineman in the same amount of time. Thus, the halfback is three times as fast (vhalfback = 9 m/s).
B. Both the halfback and the tight end have the greatest momentum. The each have the same amount of momentum - 540 kg*m/s. The lineman only has 360 kg*m/s.
4. Which case (A or B) has the greatest change in velocity, greatest acceleration, greatest momentum change, and greatest impulse.
Velocity-Time Graph
[pic]
|Greatest velocity change? |
|Greatest acceleration? |
|Greatest momentum change? |
|Greatest Impulse? |
ANSWER: a. The velocity change is greatest in case A. The v changes from +5 m/s to -3 m/s. This is a change of 8 m/s (-) and is greater than in case B (-4 m/s).
b. The acceleration is greatest in case A. Acceleration depends on velocity change and the velocity change is greatest in case A (as stated above).
c. The momentum change is greatest in case A. Momentum change depends on velocity change and the velocity change is greatest in case A (as stated above).
d. The impulse is greatest in case A. Impulse equals momentum change and the momentum change is greatest in case A (as stated above).
5. Which case (A or B) has the greatest change in velocity, greatest acceleration, greatest momentum change, and greatest impulse.
Ticker Tape Diagram
[pic]
|Greatest velocity change? |
|Greatest acceleration? |
|Greatest momentum change? |
|Greatest Impulse? |
Answer: a. The velocity change is greatest in case B. In each case the initial velocity is the same. In case B, the object rebounds in the opposite direction with a greater speed than in case A. This is equivalent to a change from +10 m/s to -5 m/s; whereas, case A has a change from +10 m/s to -2 m/s.
b. The acceleration is greatest in case B. Acceleration depends on velocity change and the velocity change is greatest in case B (as stated above)
c. The momentum change is greatest in case B. Momentum change depends on velocity change and the velocity change is greatest in case B (as stated above).
d. The impulse is greatest in case B. Impulse equals momentum change and the momentum change is greatest in case B (as stated above)
6. Use the impulse-momentum change principle to fill in the blanks in the following rows of the table. As you do, keep these three major truths in mind:
• the impulse experienced by an object is the force•time
• the momentum change of an object is the mass•velocity change
• the impulse equals the momentum change
| |Force |time |Impulse |Mom. Change |Mass |Vel. Change |
| |(N) |(s) |(N*s) |(kg*m/s) |(kg) |(m/s) |
|1. | |0.010 | | |10 |-4 |
|2. | |0.100 |-40 | |10 | |
|3. | |0.010 | |-200 |50 | |
|4. |-20 000 | | |-200 | |-8 |
|5. |-200 |1.0 | | |50 | |
Answer
| |Force |time |Impulse |Mom. Change |Mass |Vel. Change |
| |(N) |(s) |(N*s) |(kg*m/s) |(kg) |(m/s) |
|1. |-4000 |0.010 |-40 |-40 |10 |-4 |
|2. |-400 |0.100 |-40 |-40 |10 |-4 |
|3. |-20000 |0.010 |-40 |-200 |50 |-4 |
|4. |-20 000 |0.010 |-200 |-200 |25 |-8 |
|5. |-200 |1.0 |-200 |-200 |50 |-4 |
7. A 0.50-kg cart (#1) is pulled with a 1.0-N force for 1 second; another 0.50 kg cart (#2) is pulled with a 2.0 N-force for 0.50 seconds. Which cart (#1 or #2) has the greatest acceleration? Explain.
Which cart (#1 or #2) has the greatest impulse? Explain.
Which cart (#1 or #2) has the greatest change in momentum? Explain.
Answer: Cart #2 has the greatest acceleration. Recall that acceleration depends on force and mass. They each have the same mass, yet cart #2 has the greater force.
The impulse is the same for each cart. Impulse is force*time and can be calculated to be 1.0 N*s for each cart.
The momentum change is the same for each cart. Momentum change equals the impulse; if each cart has the same impulse, then it would follow that they have the same momentum change.
8. In a physics demonstration, two identical balloons (A and B) are propelled across the room on horizontal guide wires. The motion diagrams (depicting the relative position of the balloons at time intervals of 0.05 seconds) for these two balloons are shown below.
[pic]
Which balloon (A or B) has the greatest acceleration? Explain.
Which balloon (A or B) has the greatest final velocity? Explain.
Which balloon (A or B) has the greatest momentum change? Explain.
Which balloon (A or B) experiences the greatest impulse? Explain.
Answer: Balloon B has the greatest acceleration. The rate at which the velocity changes is greatest for Balloon B; this is shown by the fact that the speed (distance/time) changes most rapidly.
Balloon B has the greatest final velocity. At the end of the diagram, the distance traveled in the last interval is greatest for Balloon B.
Balloon B has the greatest momentum change. Since the final velocity is greatest for Balloon B, its velocity change is also the greatest. Momentum change depends on velocity change. The balloon with the greatest velocity change will have the greatest momentum change.
Balloon B has the greatest impulse. Impulse is equal to momentum change. If balloon B has the greatest momentum change, then it must also have the greatest impulse.
9. Two cars of equal mass are traveling down Lake Avenue with equal velocities. They both come to a stop over different lengths of time. The ticker tape patterns for each car are shown on the diagram below.
[pic]
At what approximate location on the diagram (in terms of dots) does each car begin to experience the impulse?
Which car (A or B) experiences the greatest acceleration? Explain.
Which car (A or B) experiences the greatest change in momentum? Explain.
Which car (A or B) experiences the greatest impulse? Explain.
Answer: The collision occurs at approximately the ninth dot (plus or minus a dot). The diagram shows that it is at that location that the cars begin to slow down.
Car A has the greatest acceleration. The velocity change of each car is the same. (They start with the same velocity and each finish with zero velocity.) Yet car A accomplishes this change in less time. Car A accelerates "most rapidly."
The momentum change is the same for each car. The velocity change of each car is the same (they start with the same velocity and each finish with zero velocity), and the mass of each car is the same. Thus, the momentum change is the same for each car.
The impulse is the same for each car. The impulse equals the momentum change. If the momentum change is the same for each car, then so must be the impulse.
10. The diagram to the right depicts the before- and after-collision speeds of a car which undergoes a head-on-collision with a wall. In Case A, the car bounces off the wall. In Case B, the car crumples up and sticks to the wall.
a. In which case (A or B) is the change in velocity the greatest? Explain.
b. In which case (A or B) is the change in momentum the greatest? Explain.
c. In which case (A or B) is the impulse the greatest? Explain.
d. In which case (A or B) is the force which acts upon the car the greatest (assume contact times are the same in both cases)? Explain.
Answer: Case A has the greatest velocity change. The velocity change is -9 m/s in case A and only -5 m/s in case B.
Case A has the greatest momentum change. The momentum change is dependent upon the velocity change; the object with the greatest velocity change has the greatest momentum change.
The impulse is greatest for Car A. The impulse equals the momentum change. If the momentum change is greatest for Car A, then so must be the impulse.
The impulse is greatest for Car A. The force is related to the impulse (I=F*t). The bigger impulse for Car A is attributed to the greater force upon Car A. Recall that the rebound effect is characterized by larger forces; car A is the car which rebounds.
11. Jennifer, who has a mass of 50.0 kg, is riding at 35.0 m/s in her red sports car when she must suddenly slam on the brakes to avoid hitting a deer crossing the road. She strikes the air bag, which brings her body to a stop in 0.500 s. What average force does the seat belt exert on her?
If Jennifer had not been wearing her seat belt and not had an air bag, then the windshield would have stopped her head in 0.002 s. What average force would the windshield have exerted on her?
Answer: F = (mass * velocity change)/time
F = (50 * 35) / 0.500
F = 3500 = 3.50 x 103 N
F = (mass * velocity change)/time
F = (50 * 35)/0.002
F = 875 000 N
Note that a 250-fold decrease in the time corresponds to a 250-fold increase in the force.
12. A hockey player applies an average force of 80.0 N to a 0.25 kg hockey puck for a time of 0.10 seconds. Determine the impulse experienced by the hockey puck.
Answer: Impulse = F*t = 80 N * 0.10 s
Impulse = 8 N*s
Note that not all the numbers are necessary for computing the impulse; don't "force" the value of mass into the computation.
13. If a 5.0-kg object experiences a 10.0-N force for a duration of 0.10-second, then what is the momentum change of the object?
Answer: Momentum Change = 1.0 kg*m/s
The momentum change = mass*velocity change. But since velocity change is not known another strategy must be used to find the momentum change. The strategy involves first finding the impulse (F*t = 1.0 N*s). Since impulse = momentum change, the answer is 1.0 N*s.
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.