Finding Angle for Ball to be Thrown to Catch Woman

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In summary, the conversation discusses finding the angle at which a ball should be thrown from a cliff so that a running woman can catch it. Equations are set up for the vertical and horizontal components of the ball's position, and it is determined that cos(x) must equal 6/20 in order for the woman to catch the ball. The conversation also mentions that finding the actual time value when the ball hits the ground may not be necessary for this particular exercise.
  • #1
merlinMan
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A ball is thrown from a bliff with an initial speed of 20m/s from the edge of a 45m high cliff. At the same time, a woman is running away from the cliff at a speed of 6m/s. She runs until she catches the ball. at what angle above the horizon should the ball be thrown so that she can catch ball.

I came up with a position function for the woman

[tex] V_{w} = 6t [/tex]

and set it equal to the vertical position function for the ball.



[tex] 6t= 45+ 20sin(X)t -\frac{-9.8}{2}t^2 [/tex]

I can't see where I go from here. I tried using quadratic but I didn't think I could combine the 6t and the 20sin(x)t, am I incorrect? How do I do it with 2 variables, the x and the t?
 
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  • #2
You are mixing up horizontal and vertical components of the ball's position.
1. The ball reaches the ground when its vertical position is 0 (ground level)
This gives you an equation for the time used, expressed in terms of the angle.
2. At this time, the horizontal component of the position must equal the position of the woman.
Air resistance is neglected.
 
  • #3
So . . .
1. This would be ... [tex] 0= 45+ 20sin(X)t -\frac{9.8}{2}t^2 [/tex] right?

2. [tex] 6t = 20cos(x)t [/tex] right?

I understand that . . . but I don't see where to go. In equation 2, if I divide by 6t the t's cancel out and I get [tex] \frac{20cos(x)}{6} = 0 [/tex]

Use quadratic to solve for the first equation and plug that into the 2nd?
 
  • #4
merlinMan said:
So . . .
1. This would be ... [tex] 0= 45+ 20sin(X)t -\frac{9.8}{2}t^2 [/tex] right?

2. [tex] 6t = 20cos(x)t [/tex] right?

I understand that . . . but I don't see where to go. In equation 2, if I divide by 6t the t's cancel out and I get [tex] \frac{20cos(x)}{6} = 0 [/tex]

Use quadratic to solve for the first equation and plug that into the 2nd?
It should be:
[tex] \frac{20cos(x)}{6} = 1[/tex], i.e [tex]\cos(x)=\frac{6}{20}[/tex]
As it happened, it was unnecessary in this particular exercise to find the actual time value when the ball hits the ground.
If the woman had started some distance off the cliff edge, you would have needed the particular time value.
 
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What is the basic concept behind finding the angle for a ball to be thrown to catch a woman?

The basic concept is to determine the angle at which the ball should be thrown in order to reach the woman at a certain distance and height, taking into account factors such as gravity and air resistance.

What factors need to be considered when determining the angle for a ball to be thrown to catch a woman?

The distance between the thrower and the woman, the height at which the woman is standing, the velocity of the ball, the gravitational pull, and air resistance are all important factors to consider.

How can mathematical equations be used to find the angle for a ball to be thrown to catch a woman?

By using equations such as the projectile motion equation and trigonometric functions, the angle can be calculated based on the known variables such as distance, height, and velocity.

Are there any other methods besides mathematical equations to determine the angle for a ball to be thrown to catch a woman?

Other methods such as trial and error or using a simulation software can also be used to find the angle, although they may not be as accurate as mathematical calculations.

What are some real-life applications of finding the angle for a ball to be thrown to catch a woman?

This concept can be applied in sports such as baseball or football, where players need to throw a ball to a specific location or to a moving target. It can also be useful in rescue situations, where a person needs to throw an object to someone in need of help.

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