A material point moves in a circle with a radius R at a speed v as it is necessary to change the speed of the movement

A material point moves in a circle with a radius R at a speed v as it is necessary to change the speed of the movement so that when the radius increases by 2 times, the centripetal acceleration remains the same.

From the condition of the problem it is known that a material point moves in a circle with a radius of R₁ at a speed v₁, while it experiences the action of centripetal acceleration: a₁ = v₁² / R₁, then v₁ = (a₁ ∙ R₁) ^ (1/2). With an increase in the radius by 2 times, that is, R₂ = 2 ∙ R₁, the centripetal acceleration remained the same, that is, a₁ = a₂ = a. To find out how to change the speed for this, we find the ratio v₂: v₁ = ((a ∙ R₂) / (a ​​∙ R₁)) ^ (1/2). Substitute the values ​​of physical quantities into the formula and make calculations:
v₂: v₁ = ((a ∙ 2 ∙ R₁) / (a ​​∙ R₁)) ^ (1/2), we get
v₂: v₁ = 2 ^ (1/2);
v₂ = 1.41 ∙ v₁.
Answer: To maintain acceleration, the speed must be increased 1.41 times.