Problem Statement:

A person is crossing a road with a certain velocity due north and sees a car moving towards the east. The relative velocity of the car with respect to the person is √2 times that of the velocity of the person. The angle made by the relative velocity with the east is?

Solution:

To solve this problem, we need to break down the velocities into their respective components and determine the angle between them. Let's consider the following:

- The velocity of the person, Vp = VpN (due north)
- The velocity of the car, Vc = VcE (towards the east)
- The relative velocity of the car with respect to the person, Vrel = VrelE (towards the east)

We are given that the relative velocity of the car with respect to the person is √2 times that of the velocity of the person. Mathematically, this can be represented as:

Vrel = √2 * Vp

Now, let's break down the velocities into their respective components:

Vp = VpN = VpN * i + 0 * j
Vc = 0 * i + VcE * j
Vrel = VrelE * i + 0 * j

The relative velocity can be calculated by subtracting the velocity of the person from the velocity of the car:

Vrel = Vc - Vp

Using the components, we can equate the x-components and y-components separately:

VrelE * i = 0 * i + VcE * j - VpN * i
0 = VcE - VpN

We know that Vrel = √2 * Vp, so we can substitute this value in:

√2 * Vp = VcE - VpN

Since the person is moving due north, VpN is positive. Therefore, we can rearrange the equation to solve for VcE:

VcE = VpN + √2 * Vp

Now, we have the values for VcE and VrelE. To find the angle between them, we can use the dot product formula:

Vc • Vrel = |Vc| * |Vrel| * cos(θ)

Since Vrel is in the same direction as VrelE, the dot product simplifies to:

VcE * VrelE = |Vc| * |Vrel| * cos(θ)

Substituting the values, we get:

(VpN + √2 * Vp) * VrelE = |Vc| * √2 * Vp * cos(θ)

Simplifying further:

VpN * VrelE + √2 * Vp * VrelE = |Vc| * √2 * Vp * cos(θ)

VpN * VrelE = |Vc| * √2 * Vp * cos(θ) - √2 * Vp * VrelE

VpN * VrelE = √2 * Vp * (|Vc| * cos(θ) - VrelE)

Dividing both sides by √2 * Vp:

VpN = |Vc| * cos(θ) - VrelE

Now, we can solve