We want to design a ball throwing robot arm that will throw a ball with initial velocity V at an angle 0 to the ground plane. Assuming that the robot arm is at coordinates r = 0 the ball will hit the ground at V sin 20 If where g = 9.8 is the gravitational constant. We want to ensure that the ball hits the ground at rf = 0.1730861 meters. Unfortunately, we are not free to choose the necessary Vo and 0 since the robot design constrains the speed to be a function of the angle as Vo = k(1+ cos 0) where k = 0.75 is a design parameter. a. Write down a single constraint in terms of 0 that ensures that the ball hits the ground at the given value of xf.
We want to design a ball throwing robot arm that will throw a ball with initial velocity V at an angle 0 to the ground plane. Assuming that the robot arm is at coordinates r = 0 the ball will hit the ground at V sin 20 If where g = 9.8 is the gravitational constant. We want to ensure that the ball hits the ground at rf = 0.1730861 meters. Unfortunately, we are not free to choose the necessary Vo and 0 since the robot design constrains the speed to be a function of the angle as Vo = k(1+ cos 0) where k = 0.75 is a design parameter. a. Write down a single constraint in terms of 0 that ensures that the ball hits the ground at the given value of xf.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:We want to design a ball throwing robot arm that will throw a ball with
initial velocity Vo at an angle 0 to the ground plane. Assuming that the
robot arm is at coordinates r = 0 the ball will hit the ground at
Vở sin 20
If -
where g = 9.8 is the gravitational constant.
We want to ensure that the ball hits the ground at rf
meters. Unfortunately, we are not free to choose the necessary Vo and 0
since the robot design constrains the speed to be a function of the angle as
0.1730861
Vo = k(1+ cos 0)
where k = 0.75 is a design parameter.
a. Write down a single constraint in terms of 0 that ensures that the ball
hits the ground at the given value of xf.
b. Convert the solution of 0 into a root finding problem, then solve this
problem by searching for 0 in the interval (0°, 40°).
c. Calculate the necessary Vo for the 0 you have found.
Hint: Although the 0 range is given in degrees, perform the calculations in
radians. Note that 360° = 2A radians.
%3D
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