The current of a river is 1.9 m/s. A person can swim 0.7 m/s. What is the best angle for the swimmer so that it is swept downstream the shortest distance possible? Distance of river is 120 m.
To determine the best angle for the swimmer to be swept downstream the shortest distance possible, we need to consider the relative velocities of the swimmer and the river current.
Let's assume that θ is the angle between the direction of the swimmer's motion and the direction of the river current. In this case, the swimmer's velocity can be resolved into two components: one parallel to the river current and one perpendicular to it.
The component parallel to the river current is given by V_parallel = V_swimmer * cos(θ), where V_swimmer is the swimmer's velocity (0.7 m/s) and θ is the angle.
The component perpendicular to the river current is given by V_perpendicular = V_swimmer * sin(θ), where V_swimmer is the swimmer's velocity (0.7 m/s) and θ is the angle.
The total velocity of the swimmer in the downstream direction (along the river current) is the sum of the river current velocity and the component parallel to the river current, resulting in V_downstream = V_parallel + V_current.
Given that the river current velocity is 1.9 m/s, we can substitute these values into the equation:
V_downstream = V_parallel + V_current
V_downstream = V_swimmer * cos(θ) + V_current
V_downstream = 0.7 * cos(θ) + 1.9
Now, the shortest distance across the river is a straight line from one bank to the other (the hypotenuse of a right-angled triangle). Let's call this distance D.
Using trigonometry, we can relate the distance D to the perpendicular component of the swimmer's velocity:
D = V_perpendicular * t
Since the swimmer moves at a constant velocity across the river, the time t taken to cross the river is given by:
t = D / V_swimmer * sin(θ)
Substituting the value of D into this equation:
t = (V_perpendicular * t) / V_swimmer * sin(θ)
1 = (V_perpendicular * t) / 0.7 * sin(θ)
Simplifying the equation:
0.7 * sin(θ) = V_perpendicular * t
We can substitute the value of V_perpendicular from earlier:
0.7 * sin(θ) = 0.7 * sin(θ)
Therefore, the shortest distance across the river is independent of the angle θ. Whether the swimmer moves straight across the river (θ = 0°) or at an angle, the distance traveled will be the same.
Thus, there is no best angle for the swimmer to be swept downstream the shortest distance possible. The swimmer can choose any angle they prefer.