A swimmer is swimming across a river that is exactly one mile wide, and the river current is 1.5 mph directly south. If the swimmer can swim through still water at 2.5 mph, and he swims east, perpendicular to the north-south bank, how far downstream will he be carried by the current as he swims across?
To determine how far downstream the swimmer will be carried by the current, we need to calculate the time it takes for the swimmer to cross the river and then use that time to determine how far the current carries the swimmer.
Let's calculate the time it takes for the swimmer to cross the river.
The swimmer is swimming at a speed of 2.5 mph through still water, and the river current is 1.5 mph directly south. This means the swimmer is effectively swimming at a speed of 2.5 mph - 1.5 mph = 1 mph eastward.
To cross a 1-mile wide river at a speed of 1 mph, it would take the swimmer 1 hour.
Now, we need to calculate how far the current carries the swimmer in 1 hour.
The current is flowing at a speed of 1.5 mph directly south.
Since the swimmer is swimming eastward, perpendicular to the current, the current does not affect the swimmer's eastward distance. Instead, it only affects the swimmer's downstream distance.
The downstream distance carried by the current is the product of the current speed and the time taken to cross the river, which is 1.5 mph * 1 hour = 1.5 miles.
Therefore, the swimmer will be carried 1.5 miles downstream as he swims across the river.
To determine how far downstream the swimmer will be carried by the current, we can break down the swimmer's motion into horizontal and vertical components.
1. Horizontal Component:
The swimmer is swimming east, perpendicular to the north-south bank. Since the river is one mile wide, we can use the swimmer's speed in still water, which is 2.5 mph, as the horizontal speed.
2. Vertical Component:
The river current is flowing directly south at a speed of 1.5 mph. Since the swimmer is not trying to counteract the current, we need to consider the effect of the current on the swimmer's vertical motion. In this case, the vertical component of the swimmer's motion is solely due to the river current.
To find the distance downstream, we need to calculate the time it takes for the swimmer to cross the river. Let's assume the swimmer's speed in still water, 2.5 mph, is the resultant velocity of the horizontal and vertical components.
The time it takes to cross the river can be calculated using the formula:
time = distance / speed
Distance = 1 mile (width of river)
Time = Distance / Resultant Speed
= 1 mile / 2.5 mph
= 0.4 hours
Now, since the swimmer is carried downstream by the current during this time, we need to calculate the distance he is carried vertically (downstream) by the current. The formula to calculate the distance is:
distance = speed * time
Distance Downstream = 1.5 mph * 0.4 hours
= 0.6 miles
Therefore, the swimmer will be carried 0.6 miles downstream by the current as he swims across the river.
he will take 1/2.5 = 0.4 hours to cross the river.
In that time, he will be carried 0.4 * 1.5 = 0.6 miles downstream.