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?

Step by step solution

01

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

02

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

03

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.

04

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

05

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.

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