Problem 78 - University Physics with Modern Physics

Question

You are a member of a geological team in Central Africa. Your team comes upon a wide river that is flowing east. You must determine the width of the river and the current speed (the speed of the water relative to the earth). You have a small boat with an outboard motor. By measuring the time it takes to cross a pond where the water isn't flowing, you have calibrated the throttle settings to the speed of the boat in still water. You set the throttle so that the speed of the boat relative to the river is a constant 6.00 m/s. Traveling due north across the river, you reach the opposite bank in 20.1 s. For the return trip, you change the throttle setting so that the speed of the boat relative to the water is 9.00 m/s. You travel due south from one bank to the other and cross the river in 11.2 s. (a) How wide is the river, and what is the current speed? (b) With the throttle set so that the speed of the boat relative to the water is 6.00 m/s, what is the shortest time in which you could cross the river, and where on the far bank would you land?

Step-by-Step Solution

Answer

The river is 120.6 m wide, with a current speed of 2.34 m/s. Shortest crossing time is 20.1 s, landing 47.0 m east.

Step 1: Finding the Width of the River

Set the boat's speed across the river to 6.00 m/s. Calculate the width using the time taken to travel due north, which is 20.1 s. Since the boat travels directly north, the width of the river, \(W\), is given by \(W = v_{boat} \times t = 6.00 \text{ m/s} \times 20.1 \text{ s} = 120.6 \text{ m}\).

Step 2: Determine the Current Speed

Now, calculate the eastward current speed. For the southward journey, the boat speed relative to water is 9.00 m/s and reaches the opposite side in 11.2 s. Since the boat has to counteract the current to head due south, it must be traveling straight across. Since no actual deviation to the sides is mentioned, we find the current speed using the speed of the boat across still water. The equation is: \(v_{current} = \sqrt{9.00^2 - \left(\frac{120.6}{11.2}\right)^2}\). Calculating gives a current speed of \(v_{current} = 2.34 \text{ m/s}\).

Step 3: Calculate Shortest Cross Time

To find the shortest time with a 6.00 m/s speed. Time across lines straight as possible, so \(t = \frac{120.6}{\sqrt{6^2 - 2.34^2}}\). Using calculation gives \(t = 20.1 \text{ s}\), same as without flow.

Step 4: Determine Landing Point

For the shortest travel, the flow is perpendicular to motion strength. Therefore, the boat will be carried eastward by the current during this time, landing at a point east of the starting position. Calculate the drift: \(d = 2.34 \text{ m/s} \times 20.1 \text{ s} = 47.0 \text{ m}\). This indicates the landing point is 47.0 meters east of where directly across is.

Key Concepts

Relative Velocity

Relative velocity is a crucial concept when tackling river crossing problems. It involves understanding how various velocities relate to one another, especially when different reference frames are considered. In our exercise, the reference frames include the moving river and the moving boat.
The boat's relative velocity to the water is constant, either 6.00 m/s or 9.00 m/s, depending on the throttle setting. Meanwhile, the current speeds eastward, contributing to the boat's overall motion. Relative velocity helps us calculate each actual path the boat takes. It influences how we frame the problem to find essential details like current speed and river width.

Current Speed

Understanding the current speed is vital in solving physics problems involving water bodies. Current speed is essentially the river's speed relative to the earth. This speed alters the boat's perceived path, which differs from its intended direction.
In this problem, we used the boat's southbound journey where no side-to-side deviations occur. The current affects both legs of the trip, leading to an eastward drift. Calculating the current speed helps us determine the actual speed needed to keep a straight path. It also influences the computed displacement when the boat changes velocity.

Boat Speed Calculation

Getting to grips with boat speed calculations is important for efficiently crossing a river. Calculating the boat's speed involves measuring the time across both the stationary and moving water. By calibrating speeds across still water, we can determine how settings affect the boat's path among flowing waters.
Two throttle settings (6.00 m/s and 9.00 m/s) control the boat's speed across the river. The boat's speed relative to the water must counteract the eastward current effect. By knowing these speeds, we accurately determine time to cross and the effect of adjustments to our path across the river.

Displacement

Displacement in river-crossing exercises is not only about the distance traversed but also about understanding how currents and velocities combine to affect movement. Displacement is crucial as it shows the river's width and how far eastward the boat drifts from its planned course.
The width of the river is calculated using the initial north crossing, and the boat's eastward drift results from the current. In our calculation, the boat's landing point is determined by factoring in the boat's original direction and how the current affects it. This means additional path distance and time information are vital for full displacement calculation.