physics question help please?

Question

u dont have to answer all of these, but please explain how you figured it out.

1. At what angle should a slingshot be oriented for maximum altitude? for maximum horizontal range?

2. neglecting air resistance, if u throw a ball straight up with a speed of 20m/s, how fast will it be moving wen u catch it?

3. A projectile must be moving 8km/s in order for it to match the curve of the Earth. Question: why is it important that such a satellite be above the earth's atmosphere?

Answer 1

1)
y(t)=sin(th)*v*t-g*t^2
vy(t)=sin(th)*v-g*t
and x(t)= cos(th)*v*t

If you maximum altitude, you want t in the vy to be largest when the shot has vy=0
under this condition
sin(th)*v/g=t
the only variable is th, and sin(th) is max when th= 90 degrees, or straight up.

Max range is a bit different.
Under this requirement
t is twice the time to apogee
t=2*sin(th)*v/g
x(t)=2*cos(th)*sin(th)*v^2/g

again, the only variable is th

when is cos*sin at max?
when th=45 degrees

2) 20 m/s

3) Atmosphere creates resistance

j