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Class PH1110A2007
Assignment 5
Due at 5:00pm on Friday, September 7, 2007
View Grading Details
A Wild Ride
A car in a roller coaster moves along a track that consists of a sequence of ups and downs. Let the x axis be parallel to the ground and the positive y axis point upward. In the time interval from to s, the trajectory of the car along a certain section of the track is given by , where Part A At is the roller coaster car ascending or descending? is a positive dimensionless constant.
Hint A.1 How to approach the problem The direction of motion of a particle is given by the direction of its velocity. In this particular case, you need to establish if velocity of the car points upward or downward. That can be easily determined by simply looking at the sign of the vertical component of the velocity of the car. Part A.2 Find the vertical component of the velocity of the car Find the y component of the velocity of the car, , at Hint A.2.a Velocity components Hint not displayed Express your answer in meters per second in terms of ANSWER: = Answer not displayed . .
ANSWER: Part B
ascending descending
Derive a general expression for the speed of the car. Hint B.1 How to approach the problem The speed of a particle is the magnitude of the velocity vector of the particle. Since the magnitude of a vector depends on its components, to find the speed of the car you need to know the components of the car's velocity. Hint B.2 Magnitude of a vector The magnitude of a vector , whose components are and , is given by .
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Part B.3 Find the components of the velocity of the car Find a general expression for and , that is, the x and y components of the velocity of the car. . The instantaneous velocity of the particle is Hint B.3.a The velocity vector Consider a particle moving in the xy plane with position vector defined as the time derivative of , that is, . Then, the components of the velocity vector are . Express your answers in meters per second in terms of components with a comma. ANSWER: , = and . and . Separate the velocity
Express your answer in meters per second in terms of ANSWER: =
Part C The roller coaster is designed according to safety regulations that prohibit the speed of the car from exceeding Find the maximum value of allowed by these regulations. Hint C.1 How to approach the problem To comply with the regulations, the speed of the car cannot exceed the given safety limit at any time. Thus, you need to determine what the maximum value of the speed is and impose the condition that such a value cannot be greater than the safety limit. Part C.2 Find the maximum value of the speed Given the expression found in Part B, find the maximum speed Hint C.2.a Using the calculus Recall that a function has a local maximum at if and and . , where .
of the car in terms of .
If you are trying to find the maximum value of a function over a fixed interval, you must also check whether the function is maximized at one of the endpoints of the interval. It might help to sketch the velocity versus time. Part C.2.b Find the first derivative of the speed As you found in Part B, the speed of the car is described by the following function: .
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Find an expression for the first derivative of the speed with respect to time. Express your answer as a function of ANSWER: = and .
Note that the derivative first of the speed is not necessarily equal to the magnitude of the car's acceleration. In two dimensions, to find the magnitude of the acceleration, you would first need to find the x and y components of the acceleration, then compute the magnitude of the acceleration vector. Part C.2.c Find the time at which the speed reaches its maximum value At what time does the speed reach its maximum value?
Remember to check not only the points where the derivative of the function is zero but also the endpoints of the interval to . Express your answer in seconds. ANSWER: = 2 .
Now calculate the value of the speed of the car at Express your answer in meters per second. ANSWER: = Answer not displayed
ANSWER:
= 1.66
Circular Launch
A ball is launched up a semicircular chute in such a way that at the top of the chute, just before it goes into free fall, the ball has a centripetal acceleration of magnitude 2 .
Part A How far from the bottom of the chute does the ball land? Part A.1 Speed of ball upon leaving chute
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Graphing Projectile Motion
For the motion diagram given , sketch the shape of the corresponding motion graphs in Parts A to D. Use the indicated coordinate system. One unit of time elapses between consecutive dots in the motion diagram.
Part A Construct a possible graph for x position versus time, Part A.1 Determine the initial value of Is the initial value of the x position positive, negative, or zero? ANSWER: positive negative zero graph .
Part A.2 Specify the shape of the
Does the x position change at a constant rate or a changing rate? ANSWER: constant changing
Since the x position changes at a constant rate (implying a constant x velocity), it must be represented by a graph with a constant slope. ANSWER:
View Part B Construct a possible graph for the y position versus time, Part B.1 Determine the initial value of Is the initial value of the y position positive, negative, or zero? ANSWER: positive negative zero graph .
Part B.2 Specify the shape of the
Does the y position change at a constant rate or a changing rate?
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ANSWER:
constant changing
Since the y position changes at a variable rate (implying a changing y velocity), it must be represented by a graph with a changing slope. ANSWER:
View Part C Construct a possible graph for the x velocity versus time, Part C.1 Determine the initial value of Is the initial value of the x velocity positive, negative, or zero? ANSWER: positive negative zero graph .
Part C.2 Specify the shape of the
Does the x velocity remain constant or does it change? ANSWER: It remains constant. It changes.
ANSWER:
View Part D Construct a possible graph for the y velocity versus time, Part D.1 Determine the initial value of Is the initial value of the y velocity positive, negative, or zero? ANSWER: positive negative zero graph
.
Part D.2 Specify the shape of the
Does the y velocity remain constant or does it change? ANSWER: It remains constant. It changes.
Part D.3 Specify the rate of change of
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Does the y velocity change at a constant rate or a changing rate? ANSWER: constant changing
Since the y velocity changes at a constant rate (implying a constant y acceleration), it must be represented by a graph with a constant slope. ANSWER:
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