North Allegheny School District / District Homepage
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Concepts of Physics
Mr. Kuffer
LINEAR MOTION
Vectors and Scalars
• A quantity that has only magnitude is referred to as a scalar quantity
• A quantity that has magnitude and direction is referred to as a vector quantity
– The arrow head indicates direction, while the length of the arrow indicates magnitude
Time Intervals and Displacement
• Displacement is the change in position of an object (defines the distance and direction between two positions)
– Therefore (d =df - di
• Time interval is the difference between ti and tf
– Therefore (t =tf – ti
Velocity and Acceleration
• Average Velocity
– Simply the slope of a Position vs. Time Graph
• V = (d / (t = (df – di) / (tf – ti)
• Average Acceleration
– Simply the slope of a Velocity vs. Time Graph
• a= (v / (t = (vf – vi) / (tf – ti)
List the six steps to the GUESSS Method
G - _____________ S - _____________
U - _____________ S - _____________
E - _____________
Use the GUESSS Method to solve all Problems (Show all Work)
1. An dragster releases its parachute at the end of a race, its velocity decreases from 36.0 m/s to 4.0 m/s over a 1.5 s time period . What is the average acceleration of the car?
2. A golf ball rolls up a hill toward a miniature-golf hole. Assign the direction toward the hole as being positive.
a) If the ball starts with a speed of 2 m/s and slows at a constant rate of .5 m/s2, What is its velocity after 2s?
b) If the Constant acceleration continues for 6 seconds, what will be the velocity then?
c) Describe in words and in a motion diagram the motion of the golf ball in part b.
SHORT ANSWER
3. What quantity is represented by the area under the curve of a velocity-time graph? _________________
4. If a velocity-time curve is a straight line parallel to the t-axis, what can you say about the acceleration? ______________________________
5. Sketch a velocity-time graph for an ambulance that goes 25 m/s toward the east for 100s, then 25 m/s toward the west for another 100s.
Mathematical Models of Motion Problems
Chapter 2 – Linear Motion
Name:_____________ Mr.Kuffer Date: _____
1. Light form the sun reaches the Earth in 8.3 minutes. The velocity of light is 3.0 x 108 m/s. How far is the Earth away from the sun?
2. You and a friend drive 50 km. You travel at 90 km/h; your friend travels at 95 km/h. How long will your friend wait for you at the end of the trip?
3. A cyclist maintains a constant velocity of 5.0 m/s. At time t = 0.0, the cyclist is 250.0 m from point A.
a. Plot a position vs. time graph of the cyclist’s location from point A at 10.0 second intervals for 60.0 s.
b. What is the cyclist’s position from point A at 60.0 s?
c. What is the displacement from the starting point at 60.0 s?
| |
After reading through the acceleration packet, make the following calculations. Make sure your calculations are neat, organized and units are used throughout the problem.
Type vi vf (m/s) Calculations a (m/s2)
|Rocket Dragster |0 | | | |
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| | | | | |
|Top Fuel Dragster|0 | | | |
| | | | | |
| | | | | |
| | | | | |
|2003 F1 Racecar |0 | | | |
| | | | | |
| | | | | |
| | | | | |
|2003 F1 Racecar |200 miles/hr. | | | |
|(braking) | | | | |
| | | | | |
|1997 Dodge Viper |0 | | | |
| | | | | |
| | | | | |
|1992 Ford RS200 |0 | | | |
| | | | | |
| | | | | |
|2003 z06 Corvette|0 | | | |
| | | | | |
| | | | | |
|Space Shuttle |0 | | | |
|(take-off) | | | | |
| | | | | |
|Rocket Sled |0 | | | |
| | | | | |
| | | | | |
|Human |0 | | | |
| | | | | |
| | | | | |
|Human |0 | | | |
| | | | | |
| | | | | |
|Cheetah |0 | | | |
| | | | | |
| | | | | |
Col. John P. Stapp
Acceleration Deceleration
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Determining g on an Incline
During the early part of the seventeenth century, Galileo experimentally examined the concept of acceleration. One of his goals was to learn more about freely falling objects. Unfortunately, his timing devices were not precise enough to allow him to study free fall directly. Therefore, he decided to limit the acceleration by using fluids, inclined planes, and pendulums. In this lab exercise, you will see how the acceleration of a rolling ball or cart depends on the ramp angle. Then, you will use your data to extrapolate to the acceleration on a vertical “ramp;” that is, the acceleration of a ball in free fall.
If the angle of an incline with the horizontal is small, a ball rolling down the incline moves slowly and can be easily timed. Using time and position data, it is possible to calculate the acceleration of the ball. When the angle of the incline is increased, the acceleration also increases. The acceleration is directly proportional to the sine of the incline angle, (θ ). A graph of acceleration versus sin(θ ) can be extrapolated to a point where the value of sin(θ ) is 1. When sinθ is 1, the angle of the incline is 90°. This is
measuring time, as Galileo did, you will use a Motion Detector to determine the acceleration. You will make quantitative measurements of the motion of a ball rolling down inclines of various small angles. From these measurements, you should be able to decide for yourself whether an extrapolation to large angles is valid.
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Figure 1
objectives
1. USE A MOTION DETECTOR TO MEASURE THE SPEED AND ACCELERATION OF A BALL AND A CART ROLLING DOWN AN INCLINE.
2. Determine the mathematical relationship between the angle of an incline and the acceleration of ball rolling down the ramp.
3. Determine the value of free fall acceleration, g, by extrapolating the acceleration vs. sine of track angle graph.
4. Compare the results for a ball with the results for a low-friction dynamics cart.
5. Determine if an extrapolation of the acceleration vs. sine of track angle is valid.
Materials
|COMPUTER |HARD BALL, APPROXIMATELY 8 CM DIAMETER |
|VERNIER COMPUTER INTERFACE |RUBBER BALL, SIMILAR SIZE |
|LOGGER PRO |DYNAMICS CART |
|VERNIER MOTION DETECTOR |METER STICK |
|RAMP |BOOKS |
Preliminary questions (Thought Experiments!!)
1. ONE OF THE TIMING DEVICES GALILEO USED WAS HIS PULSE. DROP A RUBBER BALL FROM A HEIGHT OF ABOUT 2 M AND TRY TO DETERMINE HOW MANY PULSE BEATS ELAPSED BEFORE IT HITS THE GROUND. WHAT WAS THE TIMING PROBLEM THAT GALILEO ENCOUNTERED?
2. Now measure the time it takes for the rubber ball to fall 2 m, using a wrist watch or wall clock. Did the results improve substantially?
3. Roll the hard ball down a ramp that makes an angle of about 10° with the horizontal. First use your pulse and then your wrist watch to measure the time of descent.
4. Do you think that during Galileo’s day it was possible to get useful data for any of these experiments? Why?
5. What should the d vs. t graph look like as the cart rolls down the incline? Sketch it!
6. What should the V vs. t graph look like as the cart rolls down the incline? Sketch it!
Procedure
1. CONNECT THE MOTION DETECTOR TO THE DIG/SONIC 1 CHANNEL OF THE INTERFACE.
2. Place a single book under one end of a 1 – 3 m long board or track so that it forms a small angle with the horizontal. Adjust the points of contact of the two ends of the incline, so that the distance, x, in Figure 1 is between 1 and 3 m.
3. Place the Motion Detector at the top of an incline. Place it so the ball will never be closer than 0.4 m.
4. Open the file “04 Determining g” from the Physics with Vernier folder.
5. Hold the hard ball on the incline about 0.5 m from the Motion Detector.
6. Click [pic] to begin collecting data; release the ball after the Motion Detector starts to click. Get your hand out of the Motion Detector path quickly. You may have to adjust the position and aim of the Motion Detector several times before you get it right. Adjust and repeat this step until you get a good run showing approximately constant slope on the velocity vs. time graph during the rolling of the ball.
7. Logger Pro can fit a straight line to a portion of your data. First indicate which portion is to be used by dragging across the graph to indicate the starting and ending times. Then click on the Linear Fit button, [pic], to perform a linear regression of the selected data. Use this tool to determine the slope of the velocity vs. time graph, using only the portion of the data for times when the ball was freely rolling. From the fitted line, find the acceleration of the ball (a.k.a… The Slope). Record the value in your data table.
8. Repeat Steps 5 – 7 two more times.
9. Measure the length of the incline, x, which is the distance between the two contact points of the ramp. See Figure 1.
10. Measure the height, h, the height of the book(s). These last two measurements will be used to determine the angle of the incline.
11. Raise the incline by placing a second book under the end. Adjust the books so that the distance, x, is the same as the previous reading.
12. Repeat Steps 5 – 10 for the new incline.
13. Repeat Steps 5 – 11 for 3, 4, and 5 books.
14. Repeat Steps 5 – 13 using a low-friction dynamics cart instead of the ball.
Data Table
|DATA USING CART |
| |O |H |X |ACCELERATION |Y |
|NUMBER OF BOOKS |HEIGHT OF BOOKS,|LENGTH OF |SIN(θ ) |trial 1 |trial 2 |trial 3 |Average acceleration|
| |H (M) |INCLINE, X | |(m/s2) |(m/s2) |(m/s2) |(m/s2) |
| | |(M) |Ο/Η | | | | |
|1 | | | | | | | |
|2 | | | | | | | |
|3 | | | | | | | |
|4 | | | | | | | |
|5 | | | | | | | |
Analysis
1. CALCULATE THE AVERAGE ACCELERATION FOR EACH HEIGHT.
2. Using trigonometry and your values of x and h in the data table, calculate the sine of the incline angle for each height. Note that x is the hypotenuse of a right triangle.
3. Plot a graph on Excel of the average acceleration (y axis) vs. sin(θ ). Carry the sin(θ ) axis out to 1 (one) to leave room for extrapolation.
4. Use the linear fit feature on Excel, and determine the slope. The slope can be used to determine the acceleration of the ball on an incline of any angle.
5. On the graph, carry the fitted line out to sin(90) = 1 on the horizontal axis, and read the value of the acceleration.[1]
6. How well does the extrapolated value agree with the accepted value of free-fall acceleration (g = 9.8 m/s2)? Solve for % Error!!!! Explain what it means (accuracy).
7. Why do you think the data for the dynamics cart resulted in an extrapolated value of g that was closer to the accepted value than the rolling ball data?
8. Discuss the validity of extrapolating the acceleration value to an angle of 90.
Extensions
1. USE THE MOTION DETECTOR TO MEASURE THE ACTUAL FREE FALL OF A BALL. COMPARE THE RESULTS OF YOUR EXTRAPOLATION WITH THE MEASUREMENT FOR FREE FALL.
2. Compare your results in this experiment with other measurements of g.
3. Investigate how the value of g varies around the world. For example, how does altitude affect the value of g? What other factors cause this acceleration to vary from place to place? How much can g vary at a school in the mountains compared to a school at sea level?
CONCEPTS OF PHYSICS
Vertical Motion: Vi = 0 m/s
Dome Worksheet
Dome Information:
Name Location Height (m) Height (ft)
|Skydome |Toronto, ON |43.3 |142 |
|Astrodome |Houston, TX |63.4 |208 |
|Kingdome |Seattle, WA |76.2 |250 |
|Superdome |New Orleans, LA |77.1 |253 |
|Georgiadome |Atlanta, GA |83.8 |275 |
|Cowboy Stadium |Dallas, TX |54.89 |180 |
|Ford Field |Detroit, MI |50.32 |165 |
Calculate how long it would take for an object dropped from the roof of each stadium to hit the ground and with what velocity would it hit in m/s, km/hr and miles/hr.
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SKYDOME @ NIGHT
Picket Fence Free Fall
We say an object is in free fall when the only force acting on it is the Earth’s gravitational force. No other forces can be acting; in particular, air resistance must be either absent or so small as to be ignored. When the object in free fall is near the surface of the earth, the gravitational force on it is nearly constant. As a result, an object in free fall accelerates downward at a constant rate. This acceleration is usually represented with the symbol g.
Physics students measure the acceleration due to gravity using a wide variety of timing methods. In this experiment, you will have the advantage of using a very precise timer connected to the computer and a Photogate. The Photogate has a beam of infrared light that travels from one side to the other. It can detect whenever this beam is blocked. You will drop a piece of clear plastic with evenly spaced black bars on it, called a Picket Fence. As the Picket Fence passes through the Photogate, the computer will measure the time from the leading edge of one bar blocking the beam until the leading edge of the next bar blocks the beam. This timing continues as all eight bars pass through the Photogate. From these measured times, the program will calculate the velocities and accelerations for this motion and graphs will be plotted.
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Figure 1
objective
6. MEASURE THE ACCELERATION OF A FREELY FALLING BODY (G) TO BETTER THAN 0.5% PRECISION USING A PICKET FENCE AND A PHOTOGATE.
Materials
|COMPUTER |VERNIER PHOTOGATE |
|VERNIER COMPUTER INTERFACE |PICKET FENCE |
|LOGGER PRO |CLAMP OR RING STAND TO SECURE PHOTOGATE |
Preliminary questions
1. INSPECT YOUR PICKET FENCE. YOU WILL BE DROPPING IT THROUGH A PHOTOGATE TO MEASURE G. THE DISTANCE, MEASURED FROM ONE EDGE OF A BLACK BAND TO THE SAME EDGE OF THE NEXT BAND, IS 5.0 CM. WHAT ADDITIONAL INFORMATION WILL YOU NEED TO DETERMINE THE AVERAGE SPEED OF THE PICKET FENCE AS IT MOVES THROUGH THE PHOTOGATE?
2. If an object is moving with constant acceleration, what is the shape of its velocity vs. time graph?
3. Does the initial velocity of an object have anything to do with its acceleration? For example, compared to dropping an object, if you throw it downward would the acceleration be different after you released it?
Procedure
1. FASTEN THE PHOTOGATE RIGIDLY TO A RING STAND SO THE ARMS EXTEND HORIZONTALLY, AS SHOWN IN FIGURE 1. THE ENTIRE LENGTH OF THE PICKET FENCE MUST BE ABLE TO FALL FREELY THROUGH THE PHOTOGATE. TO AVOID DAMAGING THE PICKET FENCE, MAKE SURE IT HAS A SOFT SURFACE (SUCH AS A CARPET) TO LAND ON.
2. Connect the Photogate to the DIG/SONIC 1 input of the Vernier computer interface.
3. Open the file “05 Picket Fence” in the Physics with Computers folder.
4. Observe the reading in the status bar of Logger Pro at the top of the screen. Block the Photogate with your hand; note that the GateState is shown as Blocked. Remove your hand and the display should change to Unblocked.
5. Click [pic] to prepare the Photogate. Hold the top of the Picket Fence and drop it through the Photogate, releasing it from your grasp completely before it enters the Photogate. Be careful when releasing the Picket Fence. It must not touch the sides of the Photogate as it falls and it needs to remain vertical. Click [pic] to end data collection.
6. Examine your graphs. The slope of a velocity vs. time graph is a measure of acceleration. If the velocity graph is approximately a straight line of constant slope, the acceleration is constant. If the acceleration of your Picket Fence appears constant, fit a straight line to your data. To do this, click on the velocity graph once to select it, then click [pic] to fit the line
y = mt + b to the data. Record the slope in the data table.
7. To establish the reliability of your slope measurement, repeat Steps 5 and 6 five more times. Do not use drops in which the Picket Fence hits or misses the Photogate. Record the slope values in the data table.
|Trial |1 |2 |3 |4 |5 |6 |
|Slope (m/s2) | | | | | | |
| |Minimum |Maximum |Average |
|Acceleration (m/s2) | | | |
| | |
|Acceleration due to gravity, g |± m/s2 |
|Precision (Relative Deviation) |% |
|Accuracy |% |
Data Table
ANALYSIS
1. FROM YOUR SIX TRIALS, DETERMINE THE MINIMUM, MAXIMUM, AND AVERAGE VALUES FOR THE ACCELERATION OF THE PICKET FENCE. RECORD THEM IN THE DATA TABLE.
2. Describe in words the shape of the position vs. time graph for the free fall.
3. Describe in words the shape of the velocity vs. time graph. How is this related to the shape of the position vs. time graph?
4. The average acceleration you determined represents a single best value, derived from all your measurements. The minimum and maximum values give an indication of how much the measurements can vary from trial to trial; that is, they indicate the precision of your measurement. One way of stating the precision is to take half of the difference between the minimum and maximum values and use the result as the uncertainty of the measurement. Express your final experimental result as the average value, the uncertainty. Round the uncertainty to just one digit and round the average value to the same decimal place.
For example, if your minimum, average and maximum values are 9.12, 9.93, and 10.84 m/s2, express your result as g = 9.9 ± 0.9 m/s2. Record your values in the data table
5. Determine Relative Deviation. Refer to the Paper Football Lab if you have forgotten how to solve for Relative Deviation (Sample calculation… no need to show all)
6. Compare your measurement to the generally accepted value of g (from a textbook or other source). Does the accepted value fall within the range of your values? If so, your experiment agrees with the accepted value. Determine the % Error.
7. Inspect your velocity graph. How would the associated acceleration vs. time graph look? Sketch your prediction on paper.
Extensions – tHOUGHT eXPERIMENTS
1. USE THE DISTANCE VS. TIME GRAPH TO EXPLAIN THE MOTION IN TERMS OF VELOCITY.
2. Would dropping the Picket Fence from higher above the Photogate change any of the parameters you measured? Try it.
3. Would throwing the Picket Fence downward, but letting go before it enters the Photogate, change any of your measurements? How about throwing the Picket Fence upward? Try performing these experiments.
4. How would adding air resistance change the results? Try adding a loop of clear tape to the upper end of the Picket Fence. Drop the modified Picket Fence through the Photogate and compare the results with your original free fall results.
5. Investigate how the value of g varies around the world. For example, how does altitude affect the value of g? What other factors cause this acceleration to vary at different locations? How much can g vary at a location in the mountains compared to a location at sea level?
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1. A car is coasting downhill at a speed of 3.0 m/s when the driver gets the engine started. After 2.5 s, the car is moving uphill at a speed of 4.5 m/s. Assuming that uphill is the positive direction, what is the car’s average acceleration?
2. A bus, traveling at 30 km/h, speeds up at a constant rate of 3.5 m/s2. What velocity does it reach 6.8 s later?
3. A race car traveling at 44 m/s slows at a constant rate to a velocity of 22m/s over 11 s. How far does it move during this time?
4. An airplane starts from rest and accelerates at a constant 3.0 m/s2 for 30.0 s before leaving the ground. How far did it move?
5. A plane travels 500 m while being accelerated uniformly from rest at the rate of 5.0 m/s2. What final velocity does it attain?
Vertical Motion Problems
Answer the following questions on a separate sheet of paper.
1. A potato gun shoots a potato vertically at 34.3 m/s. Fill out the following chart. (Use a = -10 m/s2)
t (s) Vf (m/s) a (m/s2)
|1 | | |
|2 | | |
|3 | | |
|3.43 | | |
|4 | | |
|5 | | |
|6 | | |
2. In the previous problem, how high did the potato rise?
3. What is the acceleration due to gravity in ft/s/s?
4. The roof of NASH is 45 feet tall. How fast must a potato be shot from someone on the ground in order to reach the roof? What is this speed in miles per hour?
5. How fast does a “Jugs” machine have to fire a baseball vertically in order to rise to a height of 20 m?
6. A cork from a champagne bottle comes out of the bottle at 13 m/s. How high does it rise? How long in the air?
7. John Evans has a vertical jump of 38 inches. What is his “hangtime”?
8. A fire hose fires water from a nozzle at 40 ft/s. When the hose is held vertically, how high does the water rise?
1. A stone dropped from the top of a building strikes the ground in 4.2 s. How tall is the building and how fast was the object traveling just before it struck the ground?
2. Silas McEvil pushes a safe off of a building that is 200 m tall. How much time does his victim on the ground below have to move out of the way? What is the speed of the safe when it strikes the ground? How tall is the building, approximately in feet?
3. How far will a paratrooper fall in 8.53 s? How fast will she be falling at that time?
4. An empty propane tank dropped from a hot air balloon hits the ground with a speed of 148.3 m/s. From what height was it released?
5. How long will it take an object in freefall to attain a velocity of -36.78 m/s?
6. How long would it take for a parachutist to be falling at 55 mi/hr?
7. A student drops a rock from a bridge to the water 12.43 m below. With what speed does the rock strike the water?
8. A tennis ball is dropped from 1.2 m above the ground. It rebounds to a height of 1.01 m
a. With what velocity does it hit the ground the first time?
b. With what velocity does it hit the ground the second time?
9. Niagara Falls is 55 m tall (real 180 ft tall).
c. How long would a person in a barrel be in free fall if they went over the falls?
d. With what velocity would they hit the water at the bottom?
e. Convert your answer to the above question to miles per hour.
Complete the following Free-Fall chart for a dropped object:
Time (s) ∆d (m) V (m/s) a (m/s2)
|1.00 | | |-10 |
| |3.63 | | |
| | |-12.3 | |
|2.33 | | | |
| | |-22.4 | |
| |10.78 | | |
|3.89 | | | |
Additional Practice
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NEED TO KNOW VOCABULARY TERMS
VELOCITY
INSTANTANEOUS VELOCITY
AVERAGE VELOCITY
PHYSICS
PRECISION
SCIENTIFIC NOTATION
ACCURACY
VECTOR
DISPLACEMENT
SIGNIFICANT DIGIT
SLOPE
ACCELERATION
SCALAR
DISTANCE
POSITION
COORDINATE SYSTEM
DERIVED UNIT
DEPENDENT VARIABLE
INDEPENDENT VARIABLE
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[1] Notice that extrapolating to the y value at the x = 1 point is equivalent to using the slope of the fitted line.
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HW # 7
HW # 1
HW # 2
LAB # 1
LAB # 2
Eqn: # 1
Eqn: # 3
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HW #9
HW # 8
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Velocity Problems
HW # 4
HW # 3
HW # 5
HW # 6
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