Engineering Time interval until a ball returns to its orign

AI Thread Summary
The discussion focuses on calculating the time interval until a ball returns to its origin when thrown from a balloon rising at a constant speed. A participant points out a misunderstanding in using the sine function, clarifying that the sine of 90 degrees equals 1, not approximately 0.89. There is confusion regarding the definitions of variables like initial velocity and how they relate to the balloon and the ground. The conversation emphasizes the importance of consistent variable definitions and proper mathematical notation. A hint is provided to rethink the problem using the context of an elevator scenario to find the relationship between the balloon's speed and the ball's return time.
tremain74
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Homework Statement
I have a problem that I am working on from my Engineering Dynamics Book and I want to see if I am on the right path. A person in a balloon rising with a constant velocity of 4 m/s propels a ball upward with velocity of 1.2 m/s relative to the balloon. After what time interval will the ball return to the balloon? Answer: t = 0.245 seconds.
Relevant Equations
I used the equation vy = -9.8*t + v0*sin0(sin of theta).
This is my solution below.

PXL_20241002_163129761.jpg
 
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Hi,

Do I really see a ##\sin(90)=0.89## in your picture?

(you should typeset the math in your post using ##\LaTeX ##!)

Then: The balloon goes up with 4 m/s, and the ball is thrown up with 1.2 m/s relative to the balloon.
So how can you write ##v_0=4## m/s ?

What equation are you solving (or rather: not solving, since you get a negative t) ?

##\ ##
 
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@tremain74, in addition to what @BvU has already said...

##\sin (90 \text { radians}) = 0.894## approx. You have forgotten to switch your calculator to degrees mode, so it is treating '90' as 90 radians. But in any case, you need to know that ##\sin (90^0) = 1## without using a calculator!

I can’t follow the logic of your working. In addition you have not made clear what your symbols mean, for example:

Is ##v_0##:
- the initial velocity of the ball relative to the balloon?
- the initial velocity of the ball relative to the ground?
- the velocity of the balloon?
- something else?

Is ##v_y##:
- the velocity of the ball relative to the balloon after some time, ##t##?
- the velocity of the ball relative to the ground after some time, ##t##?
- the initial velocity of the balloon relative to the ground?
- something else?

If you deliberately change the meaning of a symbol mid-working (you shouldn't!) you need to state this.

Rethink and try again. Here's a big hint:

You are in an elevator rising at a constant velocity. You throw a ball up at a speed of 1.2m/s relative to you and measure the time it takes to come back to you. Can you find the elevator's velocity from your result?
 
The balloon up speed is 4 m/s constant. Then the space up to
the point of return of ball it is 4*t.
The total space of the ball is (4+1.2)*t-9.8*t^2/2
 
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