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Ch. 2: Kinematics

Ch. 2: Kinematics. Homework: See Handout. Turn in Lab #1Wednesday. Ch. 2: Kinematics. how. Kinematics is the study of ________ things move. Ch. 2: 1-Dimensional Kinematics. Ch. 3: 2-Dimensional Kinematics Vectors and Projectile Motion. Ch. 4 and onwards: Dynamics. why.

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Ch. 2: Kinematics

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  1. Ch. 2: Kinematics Homework: See Handout. Turn in Lab #1Wednesday

  2. Ch. 2: Kinematics how Kinematics is the study of ________ things move. Ch. 2: 1-Dimensional Kinematics Ch. 3: 2-Dimensional Kinematics Vectors and Projectile Motion. Ch. 4 and onwards: Dynamics why Dynamics is the study of _______ things move.

  3. II. To study the motion of an object, certain tools are needed. These tools will describe the motion of an object through space. position The first of these tools is ___________. This specifies where an object is in space. For 1-dimensional motion, a single axis is sufficient. The x-axis shown below is an example of one such coordinate axis. The location of an object can be specified with this single axis.

  4. The next tool measures how far an object moves. There are two means to measure how far an object travels: __________ and ______________ . Distance Displacement Distance __________ is the total length of path traveled by an object. This is ______________ of the direction traveled. independent Displacement ______________ is the shortest straight line distance from start to finish of an object’s motion. This is ______________ on the direction of an object’s motion. dependent

  5. Dx The displacement is also written as ____ , and can be represented on the coordinate line below. xi x = 0 xf initial xi is the _________ position of the object. final xf is the _______ position of the object. displacement = Note: The standard unit for all length measurements is the _______ . Other units are allowed as well, such as: meter feet, inches, furlongs, kilometers, centimeters, etc.

  6. Example #1: An object moves on the number line shown below from point A (x = 1.00 meter) to B (x = 4.00 m) and then to C (x = -3.00 m). C A B x = 0 a. What are the distance and displacement of the object when it moves from point A to point B? displacement = Dx = xB – xA = 4.00 m – 1.00 m = 3.00 m The distance is also 3.00 meters. Whenever the object moves forward only, the distance and the displacement are equal.

  7. b. What are the distance and displacement of the object when it moves from point B to point C? C B 7.00 m The distance traveled is now ______ . displacement = Dx = xC – xB = (– 3.00 m) – (+4.00 m) = – 7.00 m The minus sign (-) is very important here! The minus sign indicates that the object moved towards the ______ . left

  8. c. What are the distance and displacement of the object when it moves from point A to point B to point C? C A B displacement = Dx = xC – xA = (– 3.00 m) – (+1.00 m) = – 4.00 m the distance traveled is broken into 2 parts: 3.00 m from A to B + 7.00 m from B to C = 10.00 m Remember, for distance we do not care about any ± sign. Distance is how far only. Displacement is the quantity that must include direction. This difference will eventually discussed in terms of vectors and scalars.

  9. B: Another tool measures how fast an object moves. There are two means to measure the rate of an object: __________ and ___________ . speed velocity speed distance traveled The average _______ is defined as the _________ __________ divided by the _________ _______ . elapsed time

  10. velocity displacement The average ________ , ___, is defined as the ______________ of the object divided by the _________ _______ . elapsed time Dt = the elapsed time. The standard unit for speed and velocity is the meter per second. This is written as m/s. Other units may be used as well.

  11. Interpretation of average velocity: If the position of the object is plotted as a function of time, as shown below, the average velocity is the slope of the line connecting the points (x1,t1) and (x2,t2).

  12. Example #2: An object moves on the number line shown below from point A (x = 1.00 meter) to B (x = 4.00 m) and then to C (x = -3.00 m). The motion from A to B takes 4.00 seconds and the motion from B to C takes 6.00 seconds. C A B x=0 a. What are the average speed and velocity of the object when it moves from point A to point B?

  13. b. What are the average speed and velocity of the object when it moves from point B to point C? C B

  14. c. What are the average speed and velocity of the object when it moves from point A to point B to point C? C A B Elapsed time = Dt = 4.00 s + 6.00 s = 10.00s When the velocity is positive, then it indicates the object is moving towards the ________ . When the velocity is negative, it indicates the object is moving towards the _______ . right left

  15. Example #3: Below is an example of graphing the motion of an object to determine its velocity. The position of the train is given at six different times. Use the data to determine the velocity of the train. Source: http://www.intuitive-calculus.com/definition-of-the-derivative.html

  16. Plot the data of the train’s motion on a graph:

  17. Draw a best fit line (or curve, if necessary) through the data points. For a best fit line, try to get as many data points below the line as above the line. Do not force a line through data that seems to suggest a curve of some kind!

  18. Since the plot does look very linear, we can pick two random points on the line and calculate the average velocity. Since the graph is a straight line, the slope of the line is the same at every point. Thus the instantaneous velocity is the same at every point, and equals the average velocity. The average velocity from t = 0 to t = 2.0 hours: The average velocity from t = 0 to t = 8.3 hours: The average velocity from t = 3.8 to t = 8.3 hours:

  19. rate of change The last tool measures the _______ ___ _______ of the velocity of the object. The name of this tool is the _________________ of the object. acceleration The average acceleration of the object is the change of the ______________ _________ divided by the _________ ______ . instantaneous velocity elapsed time initial Here the change of velocity is , where vi is the ______ final velocity, and vf is the ______ velocity. The standard units for the acceleration of an object are:

  20. Example #4: A drag racer accelerates from rest to 300.0 mph in a time of 4.000 seconds. Calculate the average acceleration of the drag car in m/s2. These units do not match those specified in the problem, so some conversion factors are needed.

  21. Example #5: Gomer needs to run 1600.0 meters at an average speed of 4.000 m/s to qualify for the Marines. If Gomer runs the first 1200.0 meters at a speed of 3.800 m/s, what speed must he maintain over the last 400.0 meters to qualify? To solve this problem, the times for each leg have to be figured out. The total time allowed is The time spent on the first part of the race is:

  22. The time spent on the first part of the race is: The time allowed for the second part of the race is: Thus the speed that Gomer needs to maintain for the last 400.0 meters is: Note that the speed for the last part and the speed for the first part do not mathematically average to make the average speed for the whole trip.

  23. Example #6: A woman and her dog are out for a morning run to the river, which is located 4.0 km away. The woman runs at 2.5 m/s in a straight line. The dog is unleashed and runs back and forth at 4.5 m/s between the owner and the river until she reaches the river. What is the total distance the dog runs? Solution: First find how much time it takes the woman to run: Next, find how far the dog runs in that time:

  24. Example #7: A body with an initial velocity of 8.00 m/s moves along a straight line with a constant acceleration and travels 640 meters in 40.0 seconds. For the 40.0 second interval, find (a) the average velocity, (b) the final velocity, and (c) the acceleration. Since the acceleration is constant, the average acceleration is equal to the instantaneous acceleration at all points of time. (a) Solution This equation is only valid when the acceleration is a constant! In general, it does not work, as seen in the “Gomer” example. (b) (c)

  25. Example #8: Bubba is driving his El Dorado convertible along a straight section of highway that parallels a set of railroad tracks. Bubba is traveling at 30.0 m/s when he catches up to the tail end of a train. The train is traveling at 25.0 m/s in the same direction. The train measures to be 2.20 km in length. How long will it take for Bubba to pass the front end of the train, and how far down the road will this occur? Fast solution: Bubba travels only 5.00 m/s faster than the train. Use this speed to find the time to travel 2.20 km. Distance Bubba travels:

  26. Simultaneous Equation Solution: Do not use the relative velocities, but rather the actual velocities relative to the ground (rather than one another) Distance train travels: Both t and d are unknowns. Distance Bubba travels: Note that in the same time, Bubba has to drive farther than the train drives, an additional distance equal to the length of the train.

  27. Subtract the two equations from one another: Distance Bubba travels:

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