Speedboat A negotiates a curve whose radius is 100 m. Speedboat B negotiates a curve whose radius is 221 m. Each boat experiences the same centripetal acceleration. What is the ratio vA/vB of the speeds of the boats?
v1^2/r1 = v2^2/r2
v1^2/v2^2 = r1/r2
v1/v2 = sqrt (r1/r2)
To find the ratio of the speeds of the boats, we can use the formula for centripetal acceleration:
a = v^2 / r,
where a is the centripetal acceleration, v is the velocity of the boat, and r is the radius of the curve.
Since the centripetal acceleration is the same for both boats, we can set up the following equation:
a_A = a_B.
Substituting the respective variables for each boat:
(v_A^2) / r_A = (v_B^2) / r_B.
We are given the values for the radii of the curves, r_A = 100 m and r_B = 221 m.
Inserting these values into the equation:
(v_A^2) / 100 = (v_B^2) / 221.
Now, to find the ratio v_A/v_B, we need to solve for v_A in terms of v_B.
Cross-multiplying:
(v_A^2) * 221 = (v_B^2) * 100.
Rearranging the equation to solve for v_A:
v_A^2 = (v_B^2) * 100 / 221.
Taking the square root of both sides:
v_A = sqrt((v_B^2) * 100 / 221).
Finally, to find the ratio v_A/v_B, we substitute this value into the equation:
v_A/v_B = sqrt((v_B^2) * 100 / 221) / v_B.
Simplifying the equation further:
v_A/v_B = sqrt(100 / 221).
Therefore, the ratio v_A/v_B is equal to the square root of 100/221.