A swimmer can swim in still water at a speed
of 9.47 m/s. He intends to swim directly
across a river that has a downstream current
of 4.24 m/s.
a) How many degrees from straight across
the river should he head? Let upstream be a
positive angle.
Answer in units of ◦
a. Tan A = 4.24/9.47 = 0.44773
A = 24.1o From straight.
To determine how many degrees from straight across the river the swimmer should head, we can use trigonometry.
Let's consider the velocity of the swimmer relative to the ground. This can be calculated by using the Pythagorean theorem:
(v_swimmer)^2 = (v_river)^2 + (v_downstream)^2
where
v_swimmer = velocity of the swimmer relative to the ground (unknown)
v_river = velocity of the swimmer relative to the river (9.47 m/s)
v_downstream = downstream current velocity (4.24 m/s)
Rearranging the equation, we can solve for v_swimmer:
v_swimmer = sqrt((v_river)^2 + (v_downstream)^2)
v_swimmer = sqrt((9.47 m/s)^2 + (4.24 m/s)^2)
v_swimmer = sqrt(89.6809 + 17.9776)
v_swimmer = sqrt(107.6585)
v_swimmer ≈ 10.37 m/s
Next, we can use trigonometry to find the angle between the direction the swimmer should head and straight across the river.
tan(θ) = (v_downstream)/(v_river)
θ = tan^(-1)((v_downstream)/(v_river))
θ = tan^(-1)((4.24 m/s)/(9.47 m/s))
Using a calculator, we find
θ ≈ 25.7°
Therefore, the swimmer should head approximately 25.7° upstream from straight across the river.
To find the angle at which the swimmer should head to swim directly across the river, we can use the concept of vector addition.
Let's consider the swimmer's velocity and the river's current as vectors. The swimmer's velocity is 9.47 m/s in still water, and we'll represent it with vector A. The river's current is 4.24 m/s downstream, and we'll represent it with vector B.
To swim directly across the river, the swimmer's resultant velocity (combined velocity of the swimmer and the current) should be perpendicular to the river's current.
We can use the following formula to find the angle between two vectors:
cosθ = (A • B) / (|A| • |B|)
where θ represents the angle between vectors A and B, • represents the dot product, and | | represents the magnitude (or length) of the vector.
In this case, the dot product (A • B) is equal to the product of the magnitudes (|A| • |B|) times the cosine of the angle between them.
Now, let's calculate the dot product:
(A • B) = 9.47 m/s * 4.24 m/s * cosθ
Since the swimmer is heading directly across the river, the resultant velocity should cancel out the river's current. This means that the magnitude of the resultant velocity (|resultant velocity|) is equal to the magnitude of the river's current (|B|). Hence,
|resultant velocity| = |B| = 4.24 m/s
Using the magnitudes of the vectors, our formula becomes:
(A • B) = 9.47 m/s * 4.24 m/s * cosθ = 4.24 m/s * |A| * cosθ
We can solve this equation for cosθ:
cosθ = (A • B) / (|A| • |B|) = (4.24 m/s * |A| * cosθ) / (9.47 m/s * 4.24 m/s)
cosθ can be simplified as:
cosθ = |A| / 9.47
Now, let's substitute the value of |A|:
cosθ = 9.47 m/s / 9.47
cosθ = 1
Knowing that cos(0°) = 1, we can conclude that the angle θ is 0°. This means the swimmer should head straight across the river.
Therefore, the swimmer should head directly across the river without any deviation.