A boat has a cannon on board that faces aft. The boat is capable of traveling at

19 m/s on still water. The boat is traveling North on a river that has a current flowing south at a rate of 10 m/s. Following 20m directly behind the boat is a raft that is traveling at 13 m/s relative to the water. On the raft is a clown with a hoop. The cannon fires the cannonball at 38.5 with velocity 20 m/s. At what velocity must the clown throw the hoop in order for the cannonball to travel through it? Assume the clown throws the hoop straight up.

If the elevation angle of the cannon is 38.5 degrees

then the horizontal velocity of the cannon ball relative to earth is
u = 19 - 10 - 20cos 38.5 forever
Its location relative to earth from the time of firing is
x = u t

the initial vertical velocity of the cannon ball is
Vi = 20 sin 38.5
its vertical velocity is
v = 20 sin 38.5 - 9.81 t
its height is
h = Vi t - 4.9 t^2

now the hoop
the horizontal velocity of the hoop relative to the ground is 13-10 = 3 m/s
the x position of the hoop is
xh = -20 + 3 t
the vertical speed of the hoop is
vh = Vhi - 9.81 t
the height of the hoop is
Hh = Vhi t - 4.9 t^2

now we want
x = xh
and
h = Hh

u = -6.65 m/s

u t = -20 + 3 t
20 = (3 + 6.65)t
t = 2.07 secs

how high?
h = 12.45(2.07) - 4.9(2.07)^2
= 25.8 - 21 = 4.8 meters
so
4.8 = Vih (2.07)-4.9(2.07)^2
Vih = 12.5 meters/ second
check my arithmetic !

To find the velocity at which the clown must throw the hoop in order for the cannonball to travel through it, we need to consider the relative velocities of the cannonball, the hoop, and the river current.

Let's break down the velocities involved in this scenario:
1. Boat's velocity: The boat is traveling at 19 m/s north on a river.
2. River current velocity: The current is flowing south at a rate of 10 m/s.
3. Raft's velocity: The raft is following the boat, so its velocity relative to the water will be the same as the boat, which is 19 m/s north.
4. Cannonball's velocity: The cannonball is fired with a velocity of 20 m/s at an angle of 38.5 degrees from the boat.

To find the velocity of the cannonball relative to the still water, we need to subtract the velocity of the river current:
Cannonball's velocity relative to still water = 20 m/s * cos(38.5°) = 15.997 m/s

Now, let's consider the velocity of the cannonball relative to the raft. Since the raft is following the boat, its velocity relative to the still water is 19 m/s north. Thus, the velocity of the cannonball relative to the raft is:
Cannonball's velocity relative to raft = Cannonball's velocity relative to still water - Raft's velocity = 15.997 m/s - 19 m/s = -3.003 m/s (since it's in the opposite direction)

To align the cannonball and the hoop, the clown must throw the hoop in the same direction and with the same velocity as the cannonball's relative velocity to the raft. So, the velocity at which the clown must throw the hoop is 3.003 m/s (in the direction opposite to the raft's velocity).