Question

You want to cross a straight section of a river that has a uniform current of \(5.33 \mathrm{~m} / \mathrm{s}\) and is \(127, \mathrm{~m}\) wide. Your motorboat has an engine that can generate a speed of \(17.5 \mathrm{~m} / \mathrm{s}\) for your boat. Assume that you reach top speed right away (that is, neglect the time it takes to accelerate the boat to top speed). a) If you want to go directly across the river with a \(90^{\circ}\) angle relative to the riverbank, at what angle relative to the riverbank should you point your boat? b) How long will it take to cross the river in this way? c) In which direction should you aim your boat to achieve minimum crossing time? d) What is the minimum time to cross the river? e) What is the minimum speed of your boat that will still enable you to cross the river with a \(90^{\circ}\) angle relative to the riverbank?

Short answer

Answer: The boat should maintain an angle of approximately 17.42 degrees to go directly across the river. It takes roughly 8.24 seconds to cross at this angle. The minimum time to cross the river is about 7.26 seconds. The minimum speed needed to maintain a 90-degree angle relative to the riverbank is 5.33 m/s.

Step-by-step solution

Calculate the angle to go directly across the river

To find the angle at which the boat should be pointed, we can use the equation tan(θ) = v_current / v_an, where v_current is the velocity of the current, v_an is the boat's velocity perpendicular to the current, and θ is the angle we want to find. Solving for θ, we get: θ = arctan(v_current / v_an) Here, v_current = 5.33 m/s and v_an = sqrt(17.5^2 - 5.33^2) m/s (since the boat's speed is the resultant of the boat's speed perpendicular to the river current and speed along the river current). θ = arctan(5.33 / sqrt(17.5^2 - 5.33^2)) ≈ 17.42 degrees

Calculate the time taken to cross the river

We can now find the time taken to cross the river by using the formula t = d / v_an, where d is the width of the river and v_an is the velocity of the boat perpendicular to the river current. The width of the river, d = 127 m, and v_an = sqrt(17.5^2 - 5.33^2) m/s. t = 127 / sqrt(17.5^2 - 5.33^2) ≈ 8.24 seconds

Determine the direction for minimum crossing time

To achieve the minimum crossing time, the boat should be aimed directly towards the opposite bank, perpendicular to the river current. This means the boat should be pointed at a 90-degree angle relative to the riverbank.

Calculate the minimum time to cross the river

Using the same formula as in step 2, t = d / v_an, but this time, the boat is pointed directly across the river (90-degree angle), so v_an = 17.5 m/s. t_min = 127 / 17.5 ≈ 7.26 seconds

Calculate the minimum speed needed to maintain a 90-degree angle

To maintain a 90-degree angle relative to the riverbank, the boat's velocity should be equal to the river current's velocity. Therefore, the minimum speed needed to cross the river at a 90-degree angle is 5.33 m/s.

Key concepts

Vectors and Angles

In the context of river crossing physics problems, understanding vectors and angles is crucial. Vectors are quantities that have both magnitude and direction. When dealing with a problem like crossing a river, the boat's motion can be broken down into vector components. The direction of the river's flow is a vector, and so is the speed at which the boat moves.

• Vectors help us determine the boat's resultant velocity.
• To cross directly at a 90-degree angle across the river, we must consider both the boat's engine-generated speed and the river current's speed.
• The angle at which the boat should be pointed is calculated using inverse trigonometric functions like arctan.

By pointing the boat at the correct angle perpendicular to the current, the net path taken by the boat compensates for the current, allowing for direct crossing. The concept of angles helps us visualize and compute how the boat should be adjusted to achieve this path. For example, solving for the angle using the expression \( \theta = \arctan \left( \frac{v_{\text{current}}}{v_{\text{an}}} \right) \) balances the effect of river flow.

Relative Velocity Concepts

Relative velocity is an essential concept in solving problems related to motion in flowing mediums, such as rivers. In this river crossing scenario, the relative velocity is the velocity of the boat as perceived from a stationary point on the riverbank. This includes the contribution of both the river's current and the boat's speed relative to the water.

• Relative velocity allows us to determine how fast and in what direction the boat appears to move across the river.
• It is the vector sum of the boat's actual speed and the river's current speed.
• Understanding relative velocity is key to minimizing crossing time by choosing the optimal direction.

When the boat is aimed directly across the river (90 degrees to the bank), its relative velocity ensures it moves straight across. However, if aiming to achieve the shortest crossing time, aiming directly across aligns the boat's full speed with its perpendicular movement. This means the time taken, \( t_{\text{min}} = \frac{d}{v_{\text{an}}} \), is at its lowest.

Trigonometric Functions in Physics

Trigonometric functions play a vital role in river crossing problems, allowing us to relate angles to the sides of triangles formed by velocity components. These functions help solve for unknowns like directions or resultant speeds.

• Functions like sine, cosine, and tangent link angles and sides in right triangles.
• The arctangent function is used here to calculate the angle necessary to aim the boat to counteract river current effectively.
• These calculations simplify determining the boat's direction necessary for specific outcomes, such as traveling directly across.

Trigonometry allows clear, numerical solutions by transforming vector components into scalar relationships. In this exercise, finding \( \theta = \arctan \left( \frac{v_{\text{current}}}{\sqrt{v_{\text{boat}}^2 - v_{\text{current}}^2}} \right) \) reveals the angle to point the boat to ensure a 90-degree path relative to the riverbank. Thus, trigonometric functions offer crucial tools for navigating such dynamic environments efficiently.