A lifeguard who can swim at 1.2 m/s in still water wants to reach a dock positioned perpendicularly directly across a 550 m wide river.
If the current in the river is 0.80 m/s, how long will it take the lifeguard to reach the dock?
If instead she had decided to swim in such a way that will allow her to cross the river in a minimum amount of time, where would she land relative to the dock?
Her resultant speed is perpendicular to the flow of the river.
Thus resultant velocity, using Pythagoras theorem,
V=√(1.2²-0.8²)
=√(0.8)
=0.8944 m/s
Time to cross river
=550m/0.8944 m/s
=614.9 s
=10 minutes 15 seconds, approx.
to cross as rapidly a possible, the crossing speed is exactly 1.2 m/s,
time = 550/1.2
= 458.3 s
= 7min. 38.3 sec.
To find out how long it will take the lifeguard to reach the dock, we can use the concept of relative velocity.
Step 1: Calculate the effective velocity of the lifeguard in the direction perpendicular to the river flow.
The lifeguard's velocity in the perpendicular direction can be found by subtracting the velocity of the river current from her swimming speed.
Effective velocity = Lifeguard's swimming speed - River current velocity
Effective velocity = 1.2 m/s - 0.80 m/s
Effective velocity = 0.40 m/s
Step 2: Calculate the time taken to cross the river.
The time taken can be calculated by dividing the width of the river by the effective velocity.
Time = Distance / Velocity
Time = 550 m / 0.40 m/s
Time = 1375 s
Therefore, it will take the lifeguard approximately 1375 seconds to reach the dock.
Now, let's calculate where the lifeguard would land if she had decided to swim in such a way that she crosses the river in a minimum amount of time.
Step 1: The lifeguard should swim at an angle to the river flow such that her effective velocity is directly perpendicular to the river current.
Step 2: Calculate the effective velocity in the perpendicular direction.
The effective velocity in the perpendicular direction can be found using trigonometry.
Effective velocity (perpendicular) = Lifeguard's swimming speed * sin(angle)
The angle can be calculated using the inverse sine function:
Angle = sin^(-1)(River current velocity / Lifeguard's swimming speed)
Angle = sin^(-1)(0.80 m/s / 1.2 m/s)
Angle ≈ 41.81 degrees
Step 3: Calculate the distance traveled in the perpendicular direction.
The distance can be calculated by taking the product of the effective velocity in the perpendicular direction and the time taken to cross the river.
Distance (perpendicular) = Effective velocity (perpendicular) * Time
Distance (perpendicular) ≈ 0.40 m/s * 1375 s
Distance (perpendicular) ≈ 550 m
Therefore, if the lifeguard swims in such a way that she crosses the river in a minimum amount of time, she will land directly across from the dock.
To find out how long it will take the lifeguard to reach the dock, we need to consider the motion of the lifeguard relative to the river.
Let's break down the motion into horizontal and vertical components.
Horizontal Motion:
The lifeguard needs to swim directly across the 550 m wide river. The speed of the lifeguard in still water is 1.2 m/s, but there's a current in the river with a velocity of 0.80 m/s. This current acts perpendicular to the direction of motion.
Net horizontal speed = Speed in still water - Speed of the current
= 1.2 m/s - 0.80 m/s
= 0.40 m/s
Therefore, the lifeguard will have a net horizontal speed of 0.40 m/s as she swims across the river.
Time taken to cross the river can be calculated using the formula:
Time = Distance / Speed
Time = 550 m / 0.40 m/s
Time = 1375 seconds
So, it will take the lifeguard 1375 seconds to reach the dock.
Now let's consider the scenario where the lifeguard wants to cross the river in the minimum amount of time.
To minimize the crossing time, the lifeguard should aim to swim in a direction that gives her the shortest perpendicular distance to reach the other side of the river. This can be achieved by swimming at an angle with respect to the current.
In this case, the lifeguard should swim upstream at an angle such that she is carried downstream by the current while simultaneously making progress towards the opposite bank.
As the current flows downstream at 0.80 m/s, the lifeguard can cancel out some of the downstream motion by swimming upstream at the same speed. By doing so, she will be able to move directly towards her destination, effectively reducing the time taken.
The exact point where she would land relative to the dock would depend on the angle at which she is swimming and the distance she covers while swimming upstream against the current. To determine the specific landing spot relative to the dock, additional information such as the angle at which she is swimming and the distance she traveled upstream is required.