Problem 75 - Master Resource Book in JEE Main Physics

Question

A boat is sent across a river with a velocity of \(8 \mathrm{~km} / \mathrm{h}\). If the resultant velocity of boat is \(10 \mathrm{~km} / \mathrm{h}\), then velocity of river is \(\begin{array}{lll}\text { (a) } 10 \mathrm{~km} / \mathrm{h} & \text { (b) } 8 \mathrm{~km} / \mathrm{h} & \text { (c) } 6 \mathrm{~km} / \mathrm{h}\end{array}\) (d) \(4 \mathrm{~km} / \mathrm{h}\)

Step-by-Step Solution

Answer

The velocity of the river is 6 km/h.

Step 1: Identify Known Values

We know that the velocity of the boat across the river, denoted as the static velocity, is \(8\, \text{km/h}\). The resultant velocity, which is the velocity considering both the boat's motion and the river's current, is given as \(10\, \text{km/h}\). These values will help us find the velocity of the river.

Step 2: Understand the Relationship

The resultant velocity of the boat relative to the ground is the vector sum of the boat's velocity in still water (across the river) and the river's velocity. Since these velocities are perpendicular to each other, we can use the Pythagorean theorem to relate them.

Step 3: Set Up the Equation

Let the velocity of the river be \(v_r\, \text{km/h}\). According to the Pythagorean theorem for the velocities, the relation is: \( (v_b)^2 + (v_r)^2 = (v_{res})^2 \), where \(v_b\) is the velocity of the boat across the river and \(v_{res}\) is the resultant velocity. Plugging in the known values, we get:\[ 8^2 + v_r^2 = 10^2 \]

Step 4: Solve for the River's Velocity

Simplify the equation and solve for \(v_r\):- \(8^2 = 64\)- \(10^2 = 100\)This gives us:\[ 64 + v_r^2 = 100 \]Subtract 64 from both sides:\[ v_r^2 = 36 \]Take the square root of both sides:\[ v_r = 6 \text{ km/h} \]

Step 5: Verify the Solution

Plug the calculated velocity back into the equation to ensure correctness:\[ 64 + 6^2 = 100 \]\(36\) from \(6^2\) plus \(64\) equals to \(100\), confirming our solution is correct.

Key Concepts

Vector Addition

Understanding the concept of vector addition is crucial when analyzing relative motion problems, like the one with the boat and the river. In this context, our vectors are the boat's velocity across the river and the river current's velocity.

We treat these velocities as vectors because a velocity is a direction, as much as it is a magnitude. For instance, the boat is moving across the river (one direction), and simultaneously, the river's current is carrying it downstream (a perpendicular direction). To find the resultant velocity, or how fast and in which direction the boat actually moves over the ground, we use vector addition.

• The boat's velocity in still water is one vector.
• The river's velocity forms another vector perpendicular to the first.

The combination of these vectors gives us the resultant velocity, indicating the actual path followed by the boat.

In mathematical terms, vector addition involves summing the components of two or more vectors. If two vectors are perpendicular, as in our exercise, we can use the Pythagorean theorem to find the resultant vector's magnitude.

Pythagorean Theorem

The Pythagorean theorem is a fundamental tool used in geometry and mathematics, especially when dealing with right-angled triangles. It tells us that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem is incredibly useful for calculating distances and magnitudes in vector problems.

In our problem, we use it to find unknown attributes of the triangle formed by our velocity vectors. The boat's velocity across the river and the river current velocity form the two legs of a right triangle. The resultant velocity, which is the actual speed and direction of the boat's movement over the ground, is the hypotenuse.

The equation can be written as:
\[ (v_b)^2 + (v_r)^2 = (v_{res})^2 \]
Here, \(v_b\) is the boat's velocity across the river, \(v_r\) is the river current velocity, and \(v_{res}\) is the boat's resultant velocity. By plugging in known values and solving, we can discover the unknown velocity of the river current.

River Current Velocity

River current velocity is a key variable affecting navigation across water bodies. In the context of our exercise, it's crucial in figuring out how fast the river is flowing relative to the boat's movement. This velocity influences the boat's path and impacts how quickly it reaches its destination on the opposite shore.

This exercise demonstrates how the river's velocity can be isolated and calculated by understanding and applying vector addition and the Pythagorean theorem. With the boat crossing at a speed of 8 km/h and an observed resultant speed of 10 km/h, the river's speed is the missing piece of our vector addition equation.

To find the river current velocity, we put the given velocities into the equation:
\[ 8^2 + v_r^2 = 10^2 \]
Upon solving, \(v_r = 6\) km/h, we find that the river continues to flow at 6 km/h.

• Finding the river's velocity helps in better predicting the boat's travel path.
• It allows for strategic adjustments in navigation to counter the current's effect.

Understanding this concept is essential for mastering problems involving motion in a medium like water or wind.