Initial velocity of a projectile

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The discussion revolves around calculating the initial velocity of a projectile launched at an angle above a table. Key equations include the horizontal distance and vertical motion formulas, with the need to clarify the relationship between initial and final heights. Participants emphasize the importance of resolving the horizontal and vertical components of the initial velocity using trigonometric functions. There is a focus on ensuring that the problem is approached symbolically rather than numerically, as no specific values are provided. The overall goal is to derive the initial velocity based on the given conditions of the projectile's launch and landing.
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Homework Statement



1. A projectile is launched at angle(theta) relative to the horizontal (see the figure below). The
initial height of the projectile above the surface of the table is h, and it hits the table at
distance (delta)x from the launch point. Find the launch speed vi of the projectile.

Homework Equations



(delta x)= Vi(initial velocity of x) + (delta t)
(delta y)=Vi(initial velocity of y)(delta t)-0.5g(delta t)^2
yi=yf (initial position equals final position)


The Attempt at a Solution



i think i have to some for some variable in one equation and then plug that into the other equation and solve for Vi(initial velocity)
 
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For your first equation, you probably meant
(delta x)= Vi(initial velocity of x) * (delta t)

And for clarification. . .is the initial height the same as the final height? From reading the question it seems like it is launched above the table, and then hits the table.

Remember that Vix= Vi (cos theta)
Viy= Vi (sin theta)

Is that all the info that was given? There doesn't seem to be any numbers. . .unless the answer is not supposed to be numerical. . .
 
for clarification there are not supposed to be any numbers, i just have to solve for initial velocity.

a cannon shoots a ball that lands a certain distance from the ground. when the ball is leaving the cannon its at a height of h. find the initial velocity.
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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