Problem 89 - Physics for Scientists and Engineers with Modern Physics

Question

In hot pursuit, Agent Logan of the FBI must get directly across a 1200 -m-wide river in minimum time. The river's current is \(0.80 \mathrm{~m} / \mathrm{s}\), he can row a boat at \(1.60 \mathrm{~m} / \mathrm{s}\), and he can run \(3.00 \mathrm{~m} / \mathrm{s}\). Describe the path he should take (rowing plus running along the shore) for the minimum crossing time, and determine the minimum time.

Step-by-Step Solution

Answer

Agent Logan should row directly across the river and then run downstream. Minimum time: 1095.35 s.

Step 1: Understanding the Problem

Agent Logan needs to cross a 1200 m wide river. The river flows at a speed of 0.80 m/s, and he can row the boat at 1.60 m/s relative to the water flow. Once he reaches the other side, he can run along the shore at 3.00 m/s. We must determine the path that minimizes his crossing time.

Step 2: Determine Rowing Velocity Components

The boat speed relative to the earth is the vector sum of the boat's speed relative to the water (1.60 m/s) and the river's speed (0.80 m/s). To minimize time, Agent Logan should row perpendicularly across the river. Calculate the rowing velocity component perpendicular to the river flow: \(v_{b,w} = \sqrt{1.60^2 - 0.80^2} = \sqrt{2.56 - 0.64} = 1.39 \text{ m/s}\)

Step 3: Time to Cross the River by Rowing

Since Logan needs to row perpendicularly (directly) across the river for the shortest time, use the perpendicular rowing velocity component \(v_{b,w} = 1.39 \text{ m/s}\). The width of the river is 1200 m. Calculate the time it takes to cross: \( t_{cross} = \frac{1200}{1.39} \approx 864.75 \text{ seconds} \)

Step 4: Drift Due to River Currents

During the crossing, the current will push him downstream. Calculate this drift:The lateral speed of the current is 0.80 m/s. Use the time to cross the river: \( drift = 0.80 \times 864.75 = 691.8 \text{ meters} \)

Step 5: Running Along the Shore

After crossing, Agent Logan must run the drift distance along the shore at a speed of 3.00 m/s. Calculate the time required to cover this distance:\( t_{run} = \frac{691.8}{3.00} = 230.6 \text{ seconds} \)

Step 6: Calculate Total Minimum Time

Add the rowing time and the running time to find the total minimum time:\( t_{total} = t_{cross} + t_{run} = 864.75 + 230.6 = 1095.35 \text{ seconds} \)

Key Concepts

Kinematics

Kinematics is the study of motion without considering the forces that cause it. When objects move, whether in a straight line or along a curved path, kinematics helps us understand and predict their motion. In this problem, Agent Logan's motion across a river involves calculating how his speed changes based on different factors like the river flow and his effort rowing across. Here, the focus is on determining how he moves relative to the water and the ground, and how to calculate the time taken for these movements.
Understanding kinematics is key to calculating the times involved in both rowing and running because it allows us to break down complex motions into simpler components. We start by examining Logan's speed relative to the water, his perpendicular velocity to the river's current, and how these contribute to his overall journey across the river.

Vector Addition

Vector addition is a crucial concept when calculating motion in multiple directions, such as rowing across a flowing river. It helps determine the resultant velocity by considering both the magnitude and direction of different vectors.
In this scenario, the river has a current moving in one direction while the boat moves in another. To find Agent Logan's actual path and speed across the river, we add the velocity vectors of the boat and the river flow. This is done by determining the components of each velocity.

• The velocity of the river is 0.80 m/s directed downstream.
• The velocity of the boat relative to the water is 1.60 m/s, which is perpendicular to the river.
• Using the Pythagorean theorem, we determine the effective velocity perpendicular to the flow.

By understanding how to combine these vectors, we can compute the actual path Logan takes and how rapidly he moves across the river.

Relative Velocity

Relative velocity refers to the velocity of an object as observed from a particular frame of reference. This principle is important for Agent Logan as he needs to calculate his speed concerning both the water and the riverbank.
In this exercise, relative velocity helps us understand how the boat's speed and the river current interact. The speed of the river affects Logan’s crossing path, meaning he moves differently relative to the water than he does relative to the shore.

• The river current at 0.80 m/s impacts his trajectory.
• Logan's rowing speed remains constant at 1.60 m/s relative to water.

To find how he effectively crosses the river, we consider his speed relative to the stationary earth (or shore) and factor in the river's current. This approach enables us to calculate the drift downstream and to determine his time on the river.

Time Optimization

Time optimization is the process of finding the quickest path or solution to complete a task. In logistics or problem-solving scenarios involving movement or travel, such as Agent Logan's crossing, time optimization becomes critical.
Here, the challenge is to reduce the total time spent traveling by rowing and running. To achieve this, we aim to

• Minimize the time taken to cross the river by rowing perpendicularly to reduce drift.
• Calculate the required running distance and speed to cover the drift caused by the current.
• Add the minimized times for rowing and running to find the total trip time.

By optimizing these factors, Agent Logan's travel time across and along the riverbank is minimized, efficiently combining both rowing and running for optimal pursuit speeds.