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.
To solve this problem, we need to consider the motion of the cannonball and the hoop separately.
Let's first analyze the motion of the cannonball:
1. Find the relative velocity of the river current with respect to the still water:
Vr = 10 m/s (south)
2. The boat is traveling north with a speed of 19 m/s, so the total velocity of the boat (Vb) is:
Vb = 19 m/s (north)
3. The velocity of the cannonball fired from the cannon is given as 38.5 m/s with respect to the boat. Since the cannon faces aft, the cannonball will have the same velocity but in the opposite direction of the boat:
Vcannonball = 38.5 m/s (south)
Now let's analyze the motion of the hoop on the raft:
4. The raft is traveling at 13 m/s relative to the water, which means the velocity of the raft (Vraft) is:
Vraft = 13 m/s (north)
5. The clown throws the hoop straight up, so the velocity of the hoop with respect to the raft is zero:
Vhoop-raft = 0 m/s
To find the velocity at which the clown must throw the hoop in order for the cannonball to travel through it, we need to find the velocity of the hoop with respect to the still water.
6. The velocity of the hoop with respect to the still water (Vhoop-water) can be calculated using the relative velocities:
Vhoop-water = Vhoop-raft + Vraft
= 0 m/s + 13 m/s (north)
= 13 m/s (north)
Therefore, the clown must throw the hoop with a velocity of 13 m/s in the north direction in order for the cannonball to travel through it.
To solve this problem, it is important to understand the concept of relative velocities. The velocities of objects can be added or subtracted based on their direction of motion.
Step 1: Determine the velocity of the cannonball relative to the river.
Since the boat is moving north at 19 m/s and the river current is flowing south at 10 m/s, the velocity of the cannonball relative to the river is the vector difference between these two velocities. It can be calculated as:
Velocity of cannonball relative to the river = Velocity of boat on still water - Velocity of river current
= 19 m/s - (-10 m/s) [Subtracting a southward velocity is equivalent to adding a northward velocity]
= 29 m/s (towards north)
Step 2: Determine the velocity of the cannonball relative to the raft.
Since the raft is moving at 13 m/s towards north, the velocity of the cannonball relative to the raft can be calculated by adding the velocities of the raft and the cannonball relative to the river.
Velocity of cannonball relative to the raft = Velocity of cannonball relative to the river + Velocity of raft
= 29 m/s + 13 m/s
= 42 m/s (towards north)
Step 3: Determine the velocity of the clown's hoop.
Since the cannonball travels at a velocity of 20 m/s and an angle of 38.5 degrees, the velocity of the hoop must be equal to the cannonball's velocity in the opposite direction. Therefore, the velocity of the hoop should be 20 m/s (opposite direction of the cannonball's motion).
So, to make the cannonball travel through the hoop, the clown should throw the hoop with a velocity of 20 m/s in the opposite direction of the boat's motion.
Note: The velocity of the clown's hoop is independent of the river current as the question mentions that the cannon is facing aft and not affected by the river current.