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).

Short answer

How far downstream does the boat travel during this time? Answer: It takes the motorboat approximately 7.26 seconds to cross the river, and during this time, the boat travels about 38.70 meters downstream.

Step-by-step solution

Determine the velocity components of the boat

Firstly, we need to find the velocity components of the boat i.e., the velocity component in the direction along the width of the river and the velocity component in the direction along the current of the river. Let's consider the motion perpendicular to the current, i.e., across the river, which will not be affected by the river current. Thus, the component of velocity in this direction will be equal to the top speed of the boat, i.e., \(v_{boat} = 17.5 \mathrm{m/s}\). Now, let's consider the motion parallel to the current. It is given that the current is uniformly flowing at a speed of \(v_{current} = 5.33 \mathrm{m/s}\). This component will determine how far downstream the boat drifts while crossing the river.

Calculate the time it takes to cross the river

Since the motion perpendicular to the current is unaffected by the current, we can use the boat's top speed to find the time it takes to cross the river. The width of the river is \(w = 127\mathrm{m}\). We have already determined that the component of velocity perpendicular to the current is \(v_{boat} = 17.5 \mathrm{m/s}\). Using the formula for uniformly accelerated motion: \((distance) = (speed) \times (time)\) We get, \(time = \frac{distance}{speed}\) Plugging in the given values: \(time = \frac{w}{v_{boat}} = \frac{127}{17.5} = 7.26\mathrm{s}\) (approximately) Thus, the boat takes about 7.26 seconds to cross the river.

Calculate the distance the boat travels downstream

Now that we know the time it takes the boat to cross the river (7.26 seconds), we can find how far downstream the boat drifts due to the current. In the direction along the current, we have the velocity component \(v_{current} = 5.33\mathrm{m/s}\). We can use the same formula for time and distance to determine the distance the boat travels downstream: \(distance = (speed) \times (time)\) Plugging in the values: \(distance = v_{current} \times time = 5.33 \times 7.26 = 38.70 \mathrm{m}\) (approximately) Therefore, the boat travels about 38.70 meters downstream while crossing the river.

Key concepts

Uniform Motion

When a boat attempts to cross a river with a uniform current, an understanding of uniform motion is essential. Uniform motion refers to motion at a constant speed in a straight line. In this scenario, when the boat's engine operates at a constant maximum speed and there is no significant acceleration or deceleration, we consider the boat's motion across the river to be uniform. This assumption greatly simplifies the problem because it allows us to use the basic equation of motion:

\[ distance = speed \times time \]
This formula implies that the boat will travel the same distance in every equal time interval - an essential characteristic of uniform motion.

It's important to realize that the concept of uniform motion doesn't take into account the direction of travel; it is equally applicable for any direction the boat moves, including both perpendicular to the current and along the current. For a boat crossing a river, it helps calculate how long it will take to reach the opposite shore, assuming it maintains its top speed throughout the crossing.

Velocity Components

In physics, any velocity can be broken down into components along mutually perpendicular axes. This is especially useful in our river crossing problem. Here, the boat has a velocity component along the width of the river (perpendicular to the current), and another component along the river's current (parallel to the current).

The velocity of the boat perpendicular to the river current is unchanged by the water's movement and equals the boat's top speed when directed straight across the river. However, due to the river's current, the boat also has a velocity component along the river. The resultant path of the boat is a combination of these two velocity components.

In order to calculate the actual path of the boat, information on both components and their interaction due to the current is necessary. It involves vector addition, where the boat's velocity and the river's current velocity 'add up' to define the boat's resultant path. By understanding these components, students can more easily grasp how the boat's final position is determined.

River Current

The river current plays a decisive role in our river crossing problem. It's described as the speed at which the river water flows along its course, which, in this case, is uniform, meaning it has a constant speed and direction.

A river current can complicate the simple act of crossing a river in a straight line because it tries to carry the boat downstream. To successfully reach the opposite point directly across from the starting point, the navigator must correct the boat's heading to counteract the drifting effect of the current.

The strength and direction of the river current influence the actual trajectory of the boat. By intuitively understanding this effect, students can better plan the required direction (or angle) to steer the boat in order to compensate for the current's influence and reach their desired destination across the river. The key is to recognize that the water's movement is another factor in the vehicle's velocity that must be accounted for in planning the crossing.