How Should the Rescue Boat Adjust Its Heading if the Child Swims Upstream?

[kubaanglin]

Homework Statement

Relative motion problem:
A child in danger of drowning in a river is being carried downstream by a current that has a speed of 2.4 km/h. The child is .3 km from the shore and .4 km upstream of a boat landing where a rescue boat sets out.

a) If the boat proceeds at a maximum speed of 20 km/h relative to the water, what heading should the boat take relative to the shore?
I got 36.87 degrees from positive x-axis

b)What angle does the boat's velocity with respect to the Earth make relative to the shore?
I got 41.4 degrees from positive x-axis

c) How long does it take the boat to reach the child?
I got 1.5 min (.025 hours)

d) If the child swims upstream at 1.3 km/h instead of just floating motionless in the water, how should the rescue boat change its heading to reach the child as quickly as possible?
I am having trouble with this one.

Homework Equations

None come to mind.

The Attempt at a Solution

I have been struggling with this for the past 30 minutes. I know that the angle will be just slightly less than the angle in the answer to question a, but I have tried drawing multiple triangles with the perspective of the child in mind and have not been able to figure this problem out. I have a chapter test on projectile motion, radial motion, and relative motion tomorrow and am sure there will be a problem like this on the test. Any help will be greatly appreciated.

Thank you,
Kuba

 

[Simon Bridge]

Relative motion problem:
A child in danger of drowning in a river is being carried downstream by a current that has a speed of 2.4 km/h. The child is .3 km from the shore and .4 km upstream of a boat landing where a rescue boat sets out.

a) If the boat proceeds at a maximum speed of 20 km/h relative to the water, what heading should the boat take relative to the shore?

I got 36.87 degrees from positive x-axis

Always answer the problem in the terms presented in the problem.
In this case, I am guessing that the positive x-axis is also the river bank ... and points upstream.
But you were told to set state the heading "relative to the shore" so you should actually use the word "shore" in your answer.
You should also show your working. Even when you are right, it is useful for people trying to help you tio see how you think.

b)What angle does the boat's velocity with respect to the Earth make relative to the shore?
I got 41.4 degrees from positive x-axis

c) How long does it take the boat to reach the child?
I got 1.5 min (.025 hours)

d) If the child swims upstream at 1.3 km/h instead of just floating motionless in the water, how should the rescue boat change its heading to reach the child as quickly as possible?
I am having trouble with this one.

What effect does swimming upstream have on the child's velocity wrt the shore?

 

[kubaanglin]

I have the answer key to the problem. I understand how to get the answers to a-c. I checked them with the key and they are correct. The key says that the answer to question d is 34.66°. I know the child's velocity will lower in magnitude (2.4 km/h downstream - 1.3 km/h upstream = 1.1 km/h downstream) relative to the shore. The child would be moving 1.1 km/h upstream relative to the boat. I just don't understand how to get the angle measurement of 34.66.

 

[ehild]

Work in the frame of reference of the water, as in case a.) What is the child's velocity in that frame?

 

[Simon Bridge]

How did you find the angle before?