Problem 40 - Master Resource Book in JEE Main Physics

Question

Rain is falling vertically with a speed of \(30 \mathrm{~m} / \mathrm{s} . \mathrm{A}\) woman rides a bicycle with a speed of \(10 \mathrm{~m} / \mathrm{s}\) in the north to south direction. What is the direction in which she should hold her umbrella? [NCERT] (a) \(18^{\circ}\) with vertical (b) \(18^{\circ}\) with horizontal (c) \(28^{\circ}\) with vertical (d) \(28^{\circ}\) with horizontal

Step-by-Step Solution

Answer

Option (a) \(18^{\circ}\) with vertical.

Step 1: Understand the Problem

The rain has a vertical component of velocity, and the woman moving on the bicycle creates a horizontal component. We need to find the angle at which the resultant velocity of the rain (considering both vertical and horizontal components) makes with the vertical axis. This is the direction she should hold her umbrella.

Step 2: Identify the Velocities

The rain falls vertically at a speed of \( 30 \, \text{m/s} \) and the woman travels horizontally at a speed of \( 10 \, \text{m/s} \). These velocities will act as vector components: the rain's velocity is the vertical component and the bicycle's speed is the horizontal component.

Step 3: Calculate the Resultant Velocity

Use the Pythagorean theorem to find the resultant velocity of the rain:\[v_r = \sqrt{v_{r_v}^2 + v_{r_h}^2}\]where \( v_{r_v} = 30 \, \text{m/s} \) is the vertical component and \( v_{r_h} = 10 \, \text{m/s} \) is the horizontal component.

Step 4: Plug in the Values

Using the formula from Step 3, plug in the values,\[v_r = \sqrt{30^2 + 10^2} = \sqrt{900 + 100} = \sqrt{1000} = 31.62 \, \text{m/s}\]The resultant speed of the rain relative to the woman is approximately \( 31.62 \, \text{m/s} \).

Step 5: Find the Angle with Vertical

To find the angle \( \theta \) that the resultant velocity makes with the vertical, use the formula:\[\tan\theta = \frac{v_{r_h}}{v_{r_v}}\]\( \tan\theta = \frac{10}{30} = \frac{1}{3} \).

Step 6: Calculate the Angle

Find \( \theta \) using the inverse tangent function:\[\theta = \tan^{-1}\left(\frac{1}{3}\right) \approx 18.43^{\circ}\]Rounding this to the nearest whole degree gives approximately \( 18^{\circ} \).

Step 7: Choose the Correct Option

The angle we found, \( 18^{\circ} \), is the angle with the vertical. Therefore, the answer is option (a) \(18^{\circ}\) with vertical.

Key Concepts

Rainfall Problems

Rainfall problems, such as determining the direction to hold an umbrella while riding a bicycle, often involve understanding relative motion. Consider a scenario where rain is falling vertically at a constant speed, and a cyclist is moving horizontally. The challenge is to find the angle at which the umbrella needs to be tilted to protect the cyclist from the rain. This requires understanding how the motion vectors of the rain (vertical) and the cyclist (horizontal) combine to form a resultant vector.

When dealing with these types of problems, it is important to identify all the components involved:

• The vertical component, represented by the velocity of the rain.
• The horizontal component, represented by the velocity of the cyclist.
• The resultant velocity, which combines both components to reflect how the rain appears to fall from the cyclist's perspective.

Knowing these components allows for the calculation of the necessary angle to align the umbrella. Mathematically, this involves using techniques such as the Pythagorean theorem and trigonometric functions, which are further explored in vector addition in physics.

Vector Addition in Physics

Vector addition in physics is a powerful concept used to determine the combined effect of different velocities or forces. When dealing with two-dimensional motion, vectors can represent different influences like the vertical fall of rain and the horizontal movement of a cyclist.

The two components of motion are represented as vectors:

• The vertical vector, which is the rain’s velocity pointing straight down.
• The horizontal vector, which is the cyclist's velocity moving across the ground.

Using the method of vector addition, particularly in calculating relative motion, involves finding the resultant vector. This resultant indicates the actual path of the rain as seen by the cyclist. To calculate the magnitude of the resultant vector, the Pythagorean theorem is applied, as \[ v_r = \sqrt{v_{r_v}^2 + v_{r_h}^2} \] where \( v_{r_v} \) and \( v_{r_h} \) are the vertical and horizontal components respectively.

This process of vector addition is not just limited to rain and cycling problems. It is fundamental in physics for analyzing various systems involving multiple forces or velocities acting simultaneously.

Angles with Vertical Axis

Understanding angles with the vertical axis is crucial when analyzing motion in different directions, such as those experienced in rainfall problems. The angle a resultant vector makes with the vertical axis helps in determining how objects, like an umbrella, should be oriented in moving frames of reference.

When considering the motion of rain relative to a cyclist, the angle with the vertical axis tells us how tilted the umbrella should be to adequately shield the cyclist. This involves the use of trigonometric functions: specifically, the tangent function is utilized to determine this angle when the horizontal and vertical velocities are known.

• The formula is \[ \tan \theta = \frac{v_{r_h}}{v_{r_v}} \] where \( v_{r_h} \) is the horizontal component, and \( v_{r_v} \) is the vertical component.
• The angle \( \theta \) is then found by calculating \[ \theta = \tan^{-1} \left(\frac{v_{r_h}}{v_{r_v}}\right) \]

This calculation yields the angle at which the umbrella should be held, offering insight into how different motions must be adjusted for practical situations like staying dry during a bike ride.