A boat moving at 40 km/h is crossing a river in which the current is flowing at 10 km/h. In what direction should the boat head if it is to reach a point on the other side of the river directly opposite its starting point?
the upstream component is vboat*sinTheta
sinTheta=vstream/vboat=.25
theta=....
To determine the direction the boat should head in order to reach a point directly opposite its starting point, we need to analyze the effect of the river current on the boat's motion.
Let's break down the boat's velocity into two components: the speed it would normally move without any river current and the speed due to the river current.
The boat's velocity without any river current is 40 km/h, which we can consider as the boat's "ground speed" or "true speed."
However, due to the river current flowing at 10 km/h, the boat's actual velocity will be affected. The current will exert a perpendicular force on the boat, causing it to drift downstream.
To determine the boat's velocity relative to the ground, we can use vector addition. We need to add the velocity due to the river current to the boat's actual velocity.
The magnitude of the boat's actual velocity can be calculated using the Pythagorean theorem. Let's call the boat's actual velocity "V" and the boat's true speed "B."
V = √(B^2 + C^2),
where C is the velocity of the river current.
Plugging in the values, we have:
V = √(40^2 + 10^2)
= √(1600 + 100)
= √1700
≈ 41.23 km/h
The direction of the boat's actual velocity will be different from the direction it is heading due to the river current. In order to reach a point directly opposite its starting point, the boat should head in the direction opposite to the river current.
Therefore, the boat should head upstream, or in the direction directly against the river current.
Note: The magnitude of the boat's actual velocity will vary depending on the angle between the boat's heading and the river current. The above calculation assumes that the boat's heading is perpendicular to the river current.