A swimmer wishes to swim directly across a river that has a current of 2.5m/s. He can swim 3.5m/s in a pool. Include a diagram.
a. Calculate the heading that the swimmer must use in order to swim directly across a river relative to the bank.
b. What is the ground speed (magnitude of ground velocity)?
I assume you have drawn a diagram of the vectors involved.
If straight across is zero degrees, then he must swim upstream at an angle x such that
sinx = 2.5/3.5
the ground speed is 3.5 cosx
To solve this problem, we need to use vector addition. Let's break down the problem into two components: the swimmer's velocity in still water and the velocity of the river's current.
a. To calculate the heading the swimmer must use to swim directly across the river relative to the bank, we can use the concept of vector addition. First, draw a diagram representing the situation.
----> (Swimmer's velocity in still water)
|
| ----> (River's current velocity)
Then, draw a vector representing the velocity of the swimmer relative to the ground, which we need to determine. Let's call it Vg.
-----> (Swimmer's velocity in still water - relative to the ground)
To determine the heading of Vg, we can use trigonometry. The angle between the direction of Vg and the swimmer's velocity in still water can be found using the tangent function:
tan(θ) = 2.5 m/s / 3.5 m/s
Solving for θ:
θ = arctan(2.5 m/s / 3.5 m/s)
Using a calculator, we find that θ is approximately 35.5 degrees.
Therefore, the swimmer must use a heading of 35.5 degrees relative to the bank in order to swim directly across the river.
b. To calculate the ground speed (magnitude of ground velocity), we can again use vector addition. The ground speed is the magnitude of the resulting vector obtained by adding the swimmer's velocity in still water to the velocity of the river's current.
To find the ground speed, we need to use the Pythagorean theorem:
ground speed = sqrt((3.5 m/s)^2 + (2.5 m/s)^2)
Using a calculator, we find that the ground speed is approximately 4.3 m/s.
Therefore, the swimmer's ground speed (magnitude of ground velocity) is approximately 4.3 m/s.