Difficult 2D Kinematics Question

[michaelplease]

Homework Statement

A basketball is thrown from a height of 2m. The direction of its velocity when it hits the rim of the basketball hoop is 45 degrees below the horizontal. The basketball hoop is 3m above the ground and 4.5 m in front of the shooter.

How long does the ball take to hit the rim, and what is its initial speed?

Homework Equations

Let a = x velocity, b = intial y velocity, t = time.
Kinematics equations, also |x velocity| = |final y velocity|
a=4.5/t
1=bt-4.905t^2
b-9.81t=-a
(b-9.81t)^2=b^2-19.6

The Attempt at a Solution

I know I have more equations than unknowns, but I just go in circles when trying to solve. Sometimes I get a seemingly significant step done but then another equation shows that something went wrong. Please help!

Thank you!

 

[ehild]

You have one more equation, that for the y component of velocity in terms of time.

ehild

 

[michaelplease]

Oh yes, I forgot to write that one.
vfy = b-gt

I am still stuck though trying to solve these equations (despite having so many!) Do you mind showing me how to do the algebra?

 

[ehild]

Use the equation that |vfy|=a. Is vfy positive or negative?

ehild

 

[michaelplease]

vfy is negative if we say that positive y is up.

so I combined a = -vfy and vfy = b-gt -> a = gt - b

gt - b = 4.5/t
4.5 = gt^2 - bt
add to 2nd equation listed : 5.5 = gt^2 - 4.905 t^2
t = 1.06 s

Is that right?

Then by equation 1:
a = 4.25 m/s

By equation 3:
b = 6.15 m/s

and vo = 26.2 m/s

Is that right?

 

[ehild]

a and b are correct, but vo is not. How did you get it?

ehild

 

[michaelplease]

I must have typed something wrong into my calculator.

vo = 7.48 m/s

Thanks so much for your help!

 

[ehild]

That result is correct!

ehild