Question
a student stands on the southern bank of a river that is 50m wide and has a current with a velocity of 1.1m/s [E]. the student needs to get directly across the river using their boat. if the boat's motor can propel the boat forwards with a speed of 3.8m/s, in what direction should the student point their boat to ensure they travel directly across the river to their destination?
Answers
GPT 3.5
To ensure that the student travels directly across the river, they need to aim their boat slightly upstream from their desired destination.
First, let's break down the velocities involved:
1. Velocity of the river current: 1.1 m/s [E] (east)
2. Velocity of the boat's motor: 3.8 m/s
Now we need to determine the angle at which the student should point their boat to ensure they travel directly across the river.
Let's assume the angle as θ, measured clockwise from the east.
Using vector addition, we can find the resultant velocity (v_res) of the boat:
v_res = v_boat + v_river
v_boat = 3.8 m/s at an angle of θ
v_river = 1.1 m/s [E]
To find v_res, we can resolve v_boat into its horizontal component (v_h) and vertical component (v_v):
v_h = v_boat * cos(θ)
v_v = v_boat * sin(θ)
Therefore, the horizontal component of the resultant velocity is:
v_res_h = v_h + v_river = v_boat * cos(θ) + v_river
The vertical component of the resultant velocity is:
v_res_v = v_v = v_boat * sin(θ)
To ensure that the student travels directly across the river, the vertical component (v_v) of the resultant velocity must be zero. This means that the boat should not have any vertical velocity, only horizontal velocity.
Setting v_res_v = 0, we have:
v_boat * sin(θ) = 0
Since sin(θ) = 0 when θ = 0° or θ = 180°, we have two possible solutions:
1. θ = 0°: This means the boat should point directly east, parallel to the river current.
2. θ = 180°: This means the boat should point directly west, opposite to the river current.
However, pointing the boat directly east will cause the river current to push the boat downstream, away from the desired destination. Therefore, the correct angle for the boat to ensure it travels directly across the river is θ = 180°, i.e., the boat should point directly west, opposite to the river current.
First, let's break down the velocities involved:
1. Velocity of the river current: 1.1 m/s [E] (east)
2. Velocity of the boat's motor: 3.8 m/s
Now we need to determine the angle at which the student should point their boat to ensure they travel directly across the river.
Let's assume the angle as θ, measured clockwise from the east.
Using vector addition, we can find the resultant velocity (v_res) of the boat:
v_res = v_boat + v_river
v_boat = 3.8 m/s at an angle of θ
v_river = 1.1 m/s [E]
To find v_res, we can resolve v_boat into its horizontal component (v_h) and vertical component (v_v):
v_h = v_boat * cos(θ)
v_v = v_boat * sin(θ)
Therefore, the horizontal component of the resultant velocity is:
v_res_h = v_h + v_river = v_boat * cos(θ) + v_river
The vertical component of the resultant velocity is:
v_res_v = v_v = v_boat * sin(θ)
To ensure that the student travels directly across the river, the vertical component (v_v) of the resultant velocity must be zero. This means that the boat should not have any vertical velocity, only horizontal velocity.
Setting v_res_v = 0, we have:
v_boat * sin(θ) = 0
Since sin(θ) = 0 when θ = 0° or θ = 180°, we have two possible solutions:
1. θ = 0°: This means the boat should point directly east, parallel to the river current.
2. θ = 180°: This means the boat should point directly west, opposite to the river current.
However, pointing the boat directly east will cause the river current to push the boat downstream, away from the desired destination. Therefore, the correct angle for the boat to ensure it travels directly across the river is θ = 180°, i.e., the boat should point directly west, opposite to the river current.