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?
If the river is flowing downstream at 1.5 mi/h and he can swim 2.0 mi/h then the angle he would need to swim at in order to swim straight across would be arctan(1.5/2) which is equal to 36.87 degrees. This is because you can form a right triangle with 1.5 being opposite the angle and the 2.0 as the length adjacent to the angle. With regards to what his strategy should be, I don't know since you didn't provide what the land distance would be.
When you lift a bowling ball with a force of 86 N, the ball accelerates upward with an acceleration a. If you lift with a force of 96 N, the ball's acceleration is 2a.
To determine the athlete's strategy to minimize the total time it takes to move from point A to point B, we need to consider the relative speeds and distances involved.
Let's break down the situation:
1. The athlete's swimming speed in stationary water is 2.0 mi/h.
2. The athlete's running speed is 5.0 mi/h.
3. The river flows downstream at a speed of 1.5 mi/h.
Now, let's consider the different scenarios:
1. Swimming straight across:
- In this scenario, the athlete would swim directly from A to B without accounting for the river's flow.
- The direct distance between A and B, let's call it "d," is the length of the river.
- The time it would take for the athlete to swim directly across is d/2.0 (distance divided by the swimming speed).
- However, since the river flows at 1.5 mi/h downstream, the athlete would be carried downstream and have to compensate for the additional distance.
2. Running along the riverbank:
- In this scenario, the athlete runs along the riverbank from A to B, perpendicular to the river's flow.
- The distance the athlete needs to cover is the width of the river, which is also "d."
- The time it would take for the athlete to run along the riverbank is d/5.0 (distance divided by the running speed).
- Since the athlete is running perpendicular to the river's flow, the river's velocity does not affect the time.
To find the optimal strategy for the athlete and minimize the total time, we need to compare the time it takes in each scenario.
1. Swimming straight across (considering the river's flow):
- The net velocity of the river (relative to the ground) is 1.5 mi/h downstream.
- To counteract the river's flow and swim straight across, the athlete needs to aim upstream.
- By using vector addition, we can determine that the athlete's effective swimming speed in still water is 2.0 - 1.5 = 0.5 mi/h.
- The downstream velocity helps reduce the effective swimming speed.
- The time it would take for the athlete to swim across is d/0.5 = 2d.
2. Running along the riverbank:
- The time it would take for the athlete to run along the riverbank is d/5.0 = 0.2d.
Comparing the times, we can see that it takes 2d to swim straight across and 0.2d to run along the riverbank. As the swim time is significantly longer, the athlete should choose to run along the riverbank to minimize the total time it takes to move from point A to point B.
Therefore, the athlete's strategy is to run perpendicular to the river's flow, aiming directly from point A to point B along the riverbank.