hmwrk10e.html

Homework 10: Exercises on parametric equations

Exercise 10.1 . Find parametric equations for the point P if:

(a) P travels at constant velocity from (2,3) to (11,-3) between t=0 and t=3;
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(b) P travels at constant speed 6.5 units/sec from (1,2) in the direction of (13,7);

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Exercise 10.2 . A bicycle has wheels of radius 0.4 meter. Find:

(a) the angular velocity of the wheels in rad/sec if the bicycle is traveling at 10 m/sec;
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(b) the angular velocity of the wheels in rev/sec if the bicycle is traveling at 10 m/sec;
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(d) the speed of the bicycle in km/hr if the wheels are turning at 15 rad/sec.
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Exercise 10.3.
A truck has wheels of radius 1.6 ft. Find:

(a) the speed of the truck in mph if the wheels are turning at 5 rev/sec;
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(b) the angular velocity of the wheels in rad/sec and in rev/sec if the truck is traveling at 60 mph.
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Exercise 10.4. Find a formula for the distance in feet that a car has traveled during the first t sec if its 10-inch-radius wheels:

(a) are turning at the constant rate of 24 rad/sec;
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(b) are turning at the constant rate of 9 rev/sec;
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Exercise10.5. A wheel of radius 3 is centered at the origin. Find parameteric equations for the point P on the rim if:

(a) the wheel is spinning counterclockwise at 0.5 rad/sec, and P is at (3,0) at time t=0;
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(b) the wheel is spinning clockwise at 2 rev/sec, and P is at (0,3) at time t=0;
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(d) the wheel is spinning counterclockwise at 10 rev/sec, and P is at (0,-3) at time t=0.
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Exercise 10.6 . A girl is playing on a flatcar which is moving in the positive x-direction at 10 m/sec. At time t0 she throws the ball straight upward (as viewed from the moving train) with an initial velocity 24.5 m/sec from a height 1 meter above the flatcar. From her point of view, the ball goes straight up and down, and lands on the flatcar directly below the point from which she threw it. However, an observer on the ground sees the ball travel in an arc.

Set up a coordinate system which is fixed to the earth (i.e. one which is not moving along with the train), with the y-axis pointing up and the x-axis pointing in the direction of motion of the train. Choose the origin to be the point in space which is 1 meter below the point from which the ball was released. Neglect air resistance, and take g=9.8 m/ .

(a) Find parametric equations for the motion of the ball in this fixed coordinate system.

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(b) Find the instant when the ball reaches the peak of its trajectory.

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(d) Describe in words how you would find the angle at which the ball hits the flatcar in these coordinates.
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Exercise 10.7. Suppose that an object is thrown at 70 ft/sec at an angle of below the horizontal from a window 163 ft above the ground.

(a) Express both the horizontal distance x from the window and the height y above the ground as functions of t.
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(b) Find in terms of t.
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(d) Find the velocity vector (its magnitude and angle below the horizontal) of the object at the instant when it hits the ground.

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Exercise 10.8. Suppose that a ball is thrown from ground level at v m/sec at an angle of above the horizontal. Take the origin to be the point from which the ball is thrown.

(a) Find formulas for and . The constants and should appear in your formulas.
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(b) Find the time when the ball hits the ground, expressing your answer in terms of and . There are two ways to do this: (i) Set and solve for t. (ii) Find the time when the ball reaches its peak (see Example 10.8), and double it (because it takes an equal time to fall back to the ground). Use both methods, and see that your answers agree.
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