Comprehensive mechanics excercise

[Gavroy]

Homework Statement

hi
the excercise is the following:
a man throws a non-rotating ball from a height 2m into a 3m high basketball hoop. The center of the hoop, which has a diameter of 0.5m, is 5 m in front of the man and the ball has a diameter of 0.25m.

Now the question is: If the man throws the ball directly(that means without touching anything like the plate behind the hoop or the frame of it) into the basketball hoop, in which directions and between which velocities can he choose to throw the ball to do so?

and there is a further approximation, that the velocity of the ball, when falling through the ring can be considered as constant.

Homework Equations

well the equations should not be that hard:
when you choose connecting line between the man and the basketball hoop as the x-axis, then we have

x=v_x *t
y=v_y*t-gt²/2+2.0 m
z=v_z*t

further, we have some geometric restrictions, but they are fairly subtle, like e.g. that the ball has to surpass the frame of the hoop and that he does not touch anything of it, when falling through the hoop, but all of them somehow depend on the point where the ball falls through the ring, which raises a problem.

The Attempt at a Solution

well, my problem is, that i tried to solve this problem, without using the approximation of constant velocity, because I do not know how this could help and failed, more or less because the equations became far too cumbersome. I am just looking for a promising approach.

 

[gdbb]

I think this would be easier to visualize if z were the vertical component of the problem, rather than y. So redo the equations like so:

$$\begin{align} x &= v_x t \\ y &= v_y t \\ z &= 2.0 + v_z t - \frac {g t^2}{2} \end{align}$$

We want the ball to fall into a hoop of diameter 0.5 m, which shouldn't be hard if we make a range of values that the ball can fall in.

Hint: Use the center of the ball as your reference point for landing in that range of values.

 

[Pi-Bond]

Also it might be beneficial to set your coordinate system relative to the point from where the ball is throw, i.e. denote that point by (0,0,0). I think it will definitely reduce the amount of cumbersome equations you need to deal with.

 

[Gavroy]

okay thank you, i will consider this.

but what do you think could the constant velocity approximation be good for?

 

[Pi-Bond]

Since the ball will be in the air for a short time before going (or not going) in the basket, the viscous forces in the air won't really affect the motion that much. And the constant velocity approximation also serves to reduce the complexity of the equations you are dealing with.