Question

A particle is moving along a spiral path defined by the equation r = a * e^(b * θ), where a and b are constants. Determine the velocity of the particle at a given value of θ.

Answer 1

Explanation:

v(θ) = a * b * e^(b * θ) * ω * i + a * b^2 * e^(b * θ) * ω * jj

Answer 2

Explanation:

The velocity of the particle at a given value of θ can be found by taking the derivative of the position vector with respect to time:

v(θ) = dr/dt = dr/dθ * dθ/dt = dr/dθ * ω

where ω is the angular velocity and dr/dθ is the derivative of the position vector with respect to θ.

The position vector r = a * e^(b * θ) * i + a * b * e^(b * θ) * j

Taking the derivative with respect to θ:

dr/dθ = a * b * e^(b * θ) * i + a * b^2 * e^(b * θ) * j

So the velocity of the particle is given by:

v(θ) = a * b * e^(b * θ) * ω * i + a * b^2 * e^(b * θ) * ω * j