When a motorist is driving with velocity 6i+8j the wind appears to come from the direction i. When he doubles his velocity the wind appears to come from direction i+j. Then the true velocity of the wind is???

To determine the true velocity of the wind based on the information provided, we can use vector analysis. Let's break down the problem step by step.

Understanding the Situation

The motorist's velocity is given as a vector: **v_m = 6i + 8j**. This means the motorist is moving at a speed of 6 units in the x-direction (i) and 8 units in the y-direction (j). The wind appears to come from the direction of the vector **i**, which indicates that the relative velocity of the wind with respect to the motorist is directed along the x-axis.

Relative Wind Velocity

The apparent wind direction can be found by subtracting the motorist's velocity from the true wind velocity. Let **v_w** be the true velocity of the wind. The relative wind velocity **v_r** can be expressed as:

• **v_r = v_w - v_m**

Since the wind appears to come from the direction of **i**, we can say:

• **v_r = k * i** (for some scalar k)

First Condition

From the first condition, we have:

• **v_r = v_w - (6i + 8j) = k * i**

This implies:

• **v_w - 6i - 8j = k * i**
• **v_w = (6 + k)i + 8j**

Second Condition

When the motorist doubles his velocity, the new velocity becomes **v_m' = 12i + 16j**. The wind now appears to come from the direction of **i + j**. Thus, the relative wind velocity can be expressed as:

• **v_r' = v_w - v_m' = m * (i + j)** (for some scalar m)

Setting up the equation gives us:

• **v_w - (12i + 16j) = m * (i + j)**
• **v_w = (12 + m)i + (16 + m)j**

Equating the Two Expressions for Wind Velocity

Now we have two expressions for **v_w**:

• From the first condition: **v_w = (6 + k)i + 8j**
• From the second condition: **v_w = (12 + m)i + (16 + m)j**

Setting these equal to each other gives us two equations:

• **6 + k = 12 + m**
• **8 = 16 + m**

Solving the Equations

From the second equation, we can solve for **m**:

• **m = 8 - 16 = -8**

Substituting **m** into the first equation:

• **6 + k = 12 - 8**
• **6 + k = 4**
• **k = 4 - 6 = -2**

Finding the True Wind Velocity

Now that we have **k** and **m**, we can substitute back to find the true wind velocity:

• **v_w = (6 - 2)i + 8j = 4i + 8j**

Final Result

Thus, the true velocity of the wind is:

• **v_w = 4i + 8j**

This means the wind is blowing at a speed of 4 units in the x-direction and 8 units in the y-direction. This analysis illustrates how relative motion can be understood through vector subtraction and the concept of apparent direction.