Rain drops fall vertically at a speed of `20m//s`. At what angle do they fall on the wind screen of a car moving with a velocity of `15m//`, if the wind screen velocity incined at an agle of `23^(@)` to the vertical `("cot"^(-1)4/3~~37^(@))`

Answer

To solve the problem, we need to determine the angle at which the raindrops hit the windscreen of a car that is moving horizontally. Here's a step-by-step solution:

### Step 1: Understand the velocities involved
- The raindrops are falling vertically with a speed \( V_r = 20 \, \text{m/s} \).
- The car is moving horizontally with a speed \( V_c = 15 \, \text{m/s} \).
- The windscreen is inclined at an angle of \( 23^\circ \) to the vertical.

### Step 2: Determine the effective velocity of the rain relative to the car
Since the rain is falling vertically and the car is moving horizontally, we can represent the velocities as vectors. The vertical component of the rain's velocity is \( V_r \) and the horizontal component (due to the car's motion) is \( V_c \).

### Step 3: Calculate the angle of the resultant velocity of the rain
To find the angle \( \beta \) at which the rain appears to fall relative to the car, we can use the tangent function:
\[ \tan(\beta) = \frac{V_c}{V_r} = \frac{15}{20} = \frac{3}{4} \]
Now, we can find \( \beta \) using the inverse tangent function:
\[ \beta = \tan^{-1}\left(\frac{3}{4}\right) \]
From the problem, we know that \( \cot^{-1}\left(\frac{4}{3}\right) = 37^\circ \), which means:
\[ \beta = 37^\circ \]

### Step 4: Calculate the total angle with respect to the vertical
The total angle \( \theta \) at which the rain hits the windscreen can be calculated by adding the angle of inclination of the windscreen to the angle \( \beta \):
\[ \theta = 23^\circ + 37^\circ = 60^\circ \]

### Final Answer
The angle at which the raindrops fall on the windscreen of the car is \( 60^\circ \).

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