Given:
- The transverse velocity of the particle is 2 times the radial velocity.
- The speed of the particle at any instant is 5.
- At t=0, r=1 and θ=0.

To Find:
The polar coordinates of the particle at t=1.

Explanation:

Step 1: Understanding the Problem
We are given that a particle is moving in a plane, and its transverse velocity is 2 times the radial velocity. This means that the component of velocity along the radial direction is half of the component of velocity along the transverse direction. We are also given the speed of the particle at any instant, which is 5. We need to find the polar coordinates of the particle at t=1.

Step 2: Analyzing the Problem
Since the particle is moving in a plane, we can represent its position using polar coordinates (r, θ), where r is the radial distance and θ is the angular displacement. We are given the initial position of the particle at t=0 as r=1 and θ=0.

Step 3: Finding the Velocity Components
Let's consider the velocity of the particle at any instant as v. We can resolve this velocity into two components - one along the radial direction (v_r) and the other along the transverse direction (v_θ).

Given that the transverse velocity is 2 times the radial velocity, we can write:
v_θ = 2v_r

We are also given that the speed of the particle at any instant is 5. The speed is the magnitude of the velocity vector, so we can write:
|v| = √(v_r^2 + v_θ^2) = 5

Using the relationship v_θ = 2v_r, we can substitute this into the equation for the speed to get:
√(v_r^2 + (2v_r)^2) = 5

Simplifying the equation, we get:
√(5v_r^2) = 5
v_r = 1

Therefore, the radial velocity is 1 and the transverse velocity is 2.

Step 4: Finding the Displacement
To find the polar coordinates of the particle at t=1, we need to find the radial distance and the angular displacement at t=1.

Since the radial velocity is constant, the radial distance can be found using the formula:
r = r_0 + v_r * t

Substituting the given values, we get:
r = 1 + 1 * 1 = 2

Therefore, the radial distance at t=1 is 2.

The angular displacement can be found using the formula:
θ = θ_0 + ω * t

Since the particle is initially at θ=0 and there is no angular acceleration given, the angular displacement remains zero at t=1.

Therefore, the polar coordinates of the particle at t=1 are (r, θ) = (2, 0).

Conclusion:
The polar coordinates of the particle at t=1 are (2, 0).