Wind is blowing in the north direction at speed of 2 m/s which causes the rain to fall at some angle with the vertical. With what velocity should acyclist drive so that the rain appears vertical to him:

Options

A. 2 m/s south
B. 2m/s north
C. 4 m/s west
D. 4m/s south

Answer

To solve the problem of determining the velocity at which a cyclist should drive so that the rain appears vertical to him, we can follow these steps:

### Step 1: Understand the Problem
The wind is blowing in the north direction at a speed of 2 m/s. This causes the rain to fall at an angle with respect to the vertical. We need to find the velocity of the cyclist such that the rain appears to fall vertically to him.

### Step 2: Define the Velocities
Let:
- \( V_r \) = velocity of the rain (downward)
- \( V_w = 2 \, \text{m/s} \) = velocity of the wind (northward)
- \( V_c \) = velocity of the cyclist (unknown)

### Step 3: Set Up the Coordinate System
Assume:
- The downward direction (vertical) is negative \( j \) direction.
- The northward direction (horizontal) is positive \( i \) direction.

Thus, we can express the velocities as:
- \( V_r = -V_r \, j \) (downward)
- \( V_w = V_w \, i \) (northward)

### Step 4: Relative Velocity of Rain with Respect to Cyclist
The relative velocity of the rain with respect to the cyclist can be expressed as:
\[ V_{rc} = V_r - V_c \]
Where:
- \( V_c \) is the velocity of the cyclist in the \( i \) direction.

### Step 5: Condition for Rain to Appear Vertical
For the rain to appear vertical to the cyclist, the horizontal component of the relative velocity must be zero. This means:
\[ V_{rc} = V_r - V_c \quad \text{should have no } i \text{ component.} \]
Thus, we can write:
\[ V_c = V_w \]

### Step 6: Substitute Known Values
Substituting the known value of \( V_w \):
\[ V_c = 2 \, \text{m/s} \]

### Step 7: Direction of Cyclist's Velocity
Since the wind is blowing north, the cyclist must move in the opposite direction (south) to make the rain appear vertical to him. Thus, the velocity of the cyclist is:
\[ V_c = -2 \, \text{m/s} \, \text{(southward)} \]

### Final Answer
The cyclist should drive at a speed of **2 m/s in the south direction**.

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