Problem 35 - University Physics with Modern Physics

Question

A river flows due south with a speed of 2.0 m/s. You steer a motorboat across the river; your velocity relative to the water is 4.2 m/s due east. The river is 500 m wide. (a) What is your velocity (magnitude and direction) relative to the earth? (b) How much time is required to cross the river? (c) How far south of your starting point will you reach the opposite bank?

Step-by-Step Solution

Answer

(a) Velocity: 4.66 m/s, 25.0° south of east; (b) Time: 119.05 seconds; (c) Distance south: 238.1 meters.

Step 1: Determine the Boat's Velocity Relative to Earth

The velocity of the boat relative to water is given as 4.2 m/s due east, and the velocity of the river is 2.0 m/s due south. To find the resultant velocity of the boat relative to the earth, we need to combine these two perpendicular velocities using the Pythagorean theorem. The magnitude of the resultant velocity, \( v \), is calculated as: \[ v = \sqrt{(4.2)^2 + (2.0)^2} \approx 4.66 \text{ m/s} \] To find the direction, use the arctangent function: \( \theta = \tan^{-1}\left(\frac{2.0}{4.2}\right) \approx 25.0^\circ \) south of east.

Step 2: Calculate the Time to Cross the River

The time \( t \) to cross the river can be calculated using the width of the river and the component of velocity perpendicular to the river's flow. Since the boat's speed relative to water is 4.2 m/s directly east (perpendicular to the river), the time is: \[ t = \frac{500}{4.2} \approx 119.05 \text{ seconds} \]

Step 3: Calculate How Far South the Boat will Drift

The boat's velocity downstream (due south) is 2.0 m/s. The distance drifted south \( d \) is given by the formula: \( d = \text{velocity} \times \text{time} \). Using the time calculated in the previous step: \[ d = 2.0 \times 119.05 \approx 238.1 \text{ meters} \]

Key Concepts

River Crossing

A river crossing is more than just steering from one side to the other. It involves understanding how different forces and velocities interact. When crossing a river with a flowing current, the path you take will be affected by the velocity of the water. If you steer a boat directly across, the current will push you downstream. The key is to consider both your intended path and the river's flow. This ensures that you know where you'll end up, relative to where you started. Think of it as combining two paths: the one you steer and the natural path of the river.

Resultant Velocity

Resultant velocity is the combined effect of two or more individual velocities. In river crossing problems, it refers to the velocity of the boat relative to the earth. Here, we consider the velocity of the boat and the velocity of the river. To find the resultant, these velocities need to be combined. This requires special attention when they are perpendicular, like in this case where the boat moves east (4.2 m/s) and the river moves south (2.0 m/s). Combine them using vector addition, often solved using the Pythagorean theorem.

Pythagorean Theorem

The Pythagorean theorem is a mathematical tool that helps in calculating the magnitude of the resultant velocity when two velocities are at right angles. For a right triangle where the two legs represent the known velocities (east and south), the hypotenuse gives the total velocity. The formula is:

• \[ v = \sqrt{(\text{velocity}_{east})^2 + (\text{velocity}_{south})^2} \]

This provides not just the speed, but a complete picture when combined with direction, calculated using arctan if needed.

Time Calculation

The time it takes to cross the river is crucial for planning and safety in navigation. In this problem, the time depends on how fast you move perpendicular to the river's flow. Here, the boat moves east at 4.2 m/s. The formula is:

• \[ t = \frac{\text{distance across}}{\text{velocity}_{perpendicular}} \]

For a width of 500 m, the calculation is straightforward. It allows you to determine how long it will take to reach the other bank, assuming a straight path relative to your own direction.

Drift Calculation

Drift is the distance you are carried downstream by the river's current during your journey across. Knowing how far you will drift helps in aligning your starting point with your intended landing spot. In this context, it is the southern movement due to the river's speed. The drift depends on the river's velocity and the crossing time. Use the equation:

• \[ d = \text{velocity}_{downstream} \times \text{time}_{crossing} \]

This provides you a clear expectation of where you'll end up on the opposite bank. It's essential for plotting a successful and precise crossing.