Find the angles for the basketball throw, projectile problem

[gpmattos]

1. A basketball will be launched of height h and will hit the hoop that has a height of H. The horizontal distance between the ball and the hoop is d. Given an initial velocity Vo write both possible launch angles in function of h, H, Vo and d


2.
theta=tg^-1(Vy/Vx) (1)
Vx(t)=Vo*cos(theta) (2)
Vy(t)=Vo*sen(theta)-g*t (3)
X(t)=0+Vo*cos(theta)*t (4)
Y(t)=0+Vo*sen(theta)*t-g^2/t (5)

The Attempt at a Solution

I'm lost in how to solve this problem. My main question is:

To find the angles i have to substitute both (2) and (3) in equation (1), but when i try to do that i have the "t" in the equation. if i try to use (4) to isolate t (1) will have theta and i can't isolate it. I'm clueless as how to procede.

 

[tiny-tim]

welcome to pf!

hi gpmattos! welcome to pf!

To find the angles i have to substitute both (2) and (3) in equation (1), but when i try to do that i have the "t" in the equation. if i try to use (4) to isolate t (1) will have theta and i can't isolate it. I'm clueless as how to procede.

you should have an x equation and a y equation with the same t (and the same θ)

eliminate t, and that should give you an equation in θ that you can solve …

show us how far you get ​

 

[gpmattos]

I found some information on what i needed to find the angles and i think i did it right, but i have one question, but first i'll explain my approach:

When the ball reaches its maximum height:(t1 seconds have passed)

Vy(t)=0=Vo*senb-gt==> vsenb=gt ==> t1=vsenb/g

also:

Y(t)=H-h=L=g/2(t1^2-t2^2) being t2 the time the ball reaches its destination

X(t)=Vo*cosb*(t1+t2)


We can rewrite the equation for the Y-axis:
L=g/2(t1+t2)*(2t1-(t1+t2))

from the first equation we know that t1=vsenb/g

from the second equation we know that t1+t2=d/Vo*cosb

then: L=g/2*d/Vo*cosb*((2vosenb/g)-(d/vo*cosb))


After a lot of math we obtain:

g*d^2*tg^2b-2*d*Vo^2*tgb+((L*2*vo^2)+(g*d^2))=0

and we can solve this for tgb





My main question here is, why on the second equation do i substitute t for (t1^2-t^2) and not (t1-t2)^2 . I'm sorry if this is too trivial, but i can't seem to figure this out.

 

[tiny-tim]

hi gpmattos!

(just got up :zzz:)

… When the ball reaches its maximum height:(t1 seconds have passed) …

(try using the X² and X₂ buttons just above the Reply box )

that probably works (i haven't checked your equations),

but you don't need to split the path into two …

just use one t, the time (t₂) that the ball goes through the hoop

 

[Nickie]

I would approach this problem by first identifying and defining the variables involved. In this case, h, H, Vo, and d are the given parameters, and theta is the unknown angle. I would also note that the initial velocity (Vo) is the same in both the x and y directions, and that the acceleration due to gravity (g) is acting in the y direction only.

Next, I would use the kinematic equations to relate the variables and solve for the angle theta. From equation (1), we can see that theta is equal to the inverse tangent of Vy/Vx. Using equations (2) and (3), we can substitute in the given values and solve for Vy/Vx:

Vy/Vx = (Vo*sin(theta) - g*t) / (Vo*cos(theta))

We can then use equation (4) to solve for t:

t = d / (Vo*cos(theta))

Substituting this into the equation for Vy/Vx, we get:

Vy/Vx = (Vo*sin(theta) - g*d/(Vo*cos(theta))) / (Vo*cos(theta))

Simplifying, we get:

Vy/Vx = tan(theta) - (g*d)/(Vo^2*cos^2(theta))

Now, we can substitute this into equation (1) and solve for theta:

theta = tan^-1(Vy/Vx + (g*d)/(Vo^2*cos^2(theta)))

This is a transcendental equation and does not have a closed-form solution. However, we can use numerical methods or a graphing calculator to approximate the angles. Additionally, we can see that there are two possible angles, as there is both a positive and negative solution for theta.

To summarize, as a scientist, I would approach this problem by using the given information and the kinematic equations to relate the variables and solve for the angle theta. While there may not be a closed-form solution, we can use numerical methods or technology to approximate the angles.