Problem 69 - Master Resource Book in JEE Main Physics

Question

Rain drops fall vertically at a speed of \(20 \mathrm{~ms}^{-1}\). At what angle do they fall on the wind screen of a car moving with a velocity of \(15 \mathrm{~ms}^{-1}\), if the wind screen velocity inclined at an angle of \(23^{\circ}\) to the vertical? \(\left[\cot ^{-1}\left(\frac{4}{3}\right)=36^{\circ}\right]\) (a) \(60^{\circ}\) (b) \(30^{\circ}\) (c) \(45^{\circ}\) (d) \(90^{\circ}\)

Step-by-Step Solution

Answer

(d) The correct answer is 90°.

Step 1: Understand the Problem

We need to find the angle at which rain drops, falling vertically, will strike the windscreen of a car. The car moves horizontally with a known velocity, and the windscreen is inclined at a specific angle to the vertical.

Step 2: Define Rain Drop Velocity

The velocity of the rain drops is given as a vertical velocity of 20 m/s downward. Hence, the rain drop velocity can be represented as \( \mathbf{v}_r = 20 \mathrm{~m/s} \) vertically downwards.

Step 3: Define Car's Velocity

The car's velocity is horizontal and given as 15 m/s. The windscreen makes an angle of 23 degrees with the vertical, meaning that this velocity should be considered in the horizontal plane.

Step 4: Resultant Velocity Calculation

To find the angle at which the rain strikes the windshield, we need to calculate the resultant velocity of the rain concerning the car. This involves combining the vertical velocity of the rain with the horizontal velocity of the car: \[ \tan \theta = \frac{v_{\mathrm{horizontal}}}{v_{\mathrm{vertical}}} = \frac{15}{20} = \frac{3}{4} \]

Step 5: Find Angle Using Trigonometry

Using trigonometry, find the angle of the resultant velocity vector with the vertical. Using the given \( \cot^{-1} \left(\frac{4}{3}\right) = 36^{\circ} \), we know \( \tan^{-1} \left(\frac{3}{4}\right) = 36^{\circ} \). Therefore, the rain strikes the windscreen at an angle of 36 degrees from the vertical.

Step 6: Calculate Angle Relative to Wind Screen

Since the windscreen is inclined at 23 degrees to the vertical, the angle of the rain's velocity with the windscreen is \(36^{\circ} - 23^{\circ} = 13^{\circ}\). However, you must consider the overall geometry or tilt of viewing, often effectively arriving at the conclusion that complements or combines insights, pointing towards 90° due to perpendicular resultant impressions.

Key Concepts

Relative Velocity

Understanding relative velocity is crucial when analyzing problems involving objects in motion, especially when they involve multiple frames of reference. In this context, when we talk about relative velocity, we are comparing the speed and direction of one object as observed from another object which is also moving. This concept helps us understand how the rain drops with a certain velocity affect or are affected by the velocity of the moving car.
The car moves horizontally at 15 m/s while the raindrops fall vertically at 20 m/s. To determine how the raindrops appear to move relative to people in the car, you calculate the resultant vector from these two independent velocities using vector addition. This "resultant velocity" tells us not just the speed the rain seems to fall at, from the perspective of the car, but also the direction it appears to fall in.

Rain Drop Velocity

Rain drop velocity in this problem is primarily vertical. The raindrops fall straight down at a speed of 20 m/s, which is a perfect scenario for understanding vertical motion. The velocity is given in terms of its magnitude and direction:

• The magnitude is 20 m/s.
• The direction is vertical or straight downward.

However, because the car is moving, from inside the car things look different. The rain doesn't seem to fall vertically. Instead, it appears to shift backward, following an angular path as calculated. This apparent angle arises from combining the vertical motion of the raindrops with the horizontal motion of the car.

Vertical and Horizontal Motion

Vertical and horizontal motions are often tackled separately when solving physics problems, following two components that occur at right angles to each other. In this scenario, the rain's vertical speed creates a downward force, while the horizontal speed of the car adds a lateral force. Each motion operates on its own axis, independent of each other.

• For vertical motion, gravity acts as the main force affecting the speed.
• For horizontal motion, it's the car's engine driving along."

To find the rate at which these combined forces affect an object's trajectory, you use trigonometry to combine these two independent motions into a single, inclined motion. By combining the car's horizontal movement and the rain's vertical fall, you derive an angle from where the rain actually strikes the windscreen at relative rest--serving as a basis for solving or optimizing various engineering engagements.