An athlete in a competition needs to get from point A to another point B directly across from a river. He can swim in stationary water at a speed of 2.0 mi/h, and he can run at a speed of 5.0 mi/h. If the river does not flow, then to get from A to B he would certainly choose to swim directly across. But the river flows at a speed of 1.5 mi/h downstream. Given that, what would his strategy be in order to minimize the total time it takes to move from A to B? i.e., at what angle upstream (measured from the line AB) should he be swimming?
To minimize the total time it takes to move from point A to point B, the athlete should use a strategy called the "Minimum-Time Strategy" or "Minimum-Time Path."
To determine the angle at which he should swim upstream, we can use the concept of "vector addition" to calculate the resultant velocity.
Let's define the variables:
- V_swim: Speed of the athlete while swimming (2.0 mi/h)
- V_run: Speed of the athlete while running on land (5.0 mi/h)
- V_river: Speed of the river flow (1.5 mi/h)
- θ: Angle between the swim direction and the line AB
To find the minimum-time strategy, we need to find the angle θ that would minimize the total time it takes to cross the river.
First, let's break down the athlete's velocity into its components.
- Swim component: V_swim in the direction perpendicular to the river flow (upstream)
- River component: V_river in the direction of the river flow (downstream)
- Net velocity: The sum of the swim and river components
Now, we can calculate the net velocity. The net velocity can be found by vector addition of the swim and river components.
Using trigonometry, we know that the swim component is given by V_swim * cos(θ), and the river component is given by V_river.
Thus, the net velocity, V_net, is given by V_swim * cos(θ) + V_river.
To find the time it takes to cross the river, we need to divide the distance across the river by the net velocity.
Let's assume the width of the river is D.
The time taken to cross the river is given by T = D / (V_swim * cos(θ) + V_river).
To minimize T, we need to find the value of θ that minimizes this expression.
Differentiating T with respect to θ and setting it to zero will give us the minimum time.
d(T)/d(θ) = -D * V_swim * sin(θ) / (V_swim * cos(θ) + V_river)^2
Setting d(T)/d(θ) = 0, we get:
-D * V_swim * sin(θ) / (V_swim * cos(θ) + V_river)^2 = 0
This equation can be simplified to:
V_swim * sin(θ) = 0
Since V_swim cannot be zero, this equation gives us the only solution:
sin(θ) = 0
This implies that θ = 0 degrees.
Therefore, the athlete should swim directly across the river, at an angle of 0 degrees upstream, to minimize the total time it takes to cross from point A to point B.
To minimize the total time it takes to move from point A to point B, the athlete needs to consider the effect of the river current on his speed. Let's assume the width of the river is d.
If the athlete chooses to swim directly across the river, he will have a speed of 2.0 mi/h relative to the stationary water. However, due to the river current flowing downstream at 1.5 mi/h, his effective speed across the river (perpendicular to the current) will be reduced.
To determine the time it takes for the athlete to cross the river, we need to consider both the vertical distance he needs to cross and the horizontal distance he is displaced downstream by the current.
Let the angle between the line AB and his swimming path be θ (measured upstream). The vertical speed of the athlete (perpendicular to the current) can be determined by sin(θ) * 2.0 mi/h, and the horizontal speed (displacement by the current) can be determined by cos(θ) * 1.5 mi/h.
The time it takes to cross the river can be calculated as d divided by the vertical component of the swimming speed (sin(θ) * 2.0 mi/h). Additionally, the time it takes to be displaced downstream can be calculated by d divided by the horizontal component of the swimming speed (cos(θ) * 1.5 mi/h).
Since these two times are occurring simultaneously, the total time can be represented as:
t_Total = t_Vertical + t_Horizontal = d / (sin(θ) * 2.0) + d / (cos(θ) * 1.5)
To minimize the total time, the athlete needs to find the angle θ that minimizes this expression.
To solve for the minimum, we first need to take the derivative of the total time expression with respect to θ and set it equal to zero:
d(t_Total) / dθ = (2.0d cos(θ) - 1.5d sin(θ)) / (sin^2(θ) * cos(θ)) = 0
Simplifying this equation gives us:
2.0 cos(θ) - 1.5 sin(θ) = 0
To solve this equation, we can divide both sides by cos(θ) and rearrange:
2.0 / 1.5 = sin(θ) / cos(θ)
Using the identity tan(θ) = sin(θ) / cos(θ), we have:
tan(θ) = 2.0 / 1.5 = 4/3
Taking the inverse tangent of both sides gives us:
θ = arctan(4/3)
Therefore, the athlete should choose to swim at an angle of arctan(4/3) upstream from the line AB in order to minimize the total time it takes to move from point A to point B.