Speedboat (A) negotiates a curve whose radius is 115 m. Speedboat (B) negotiates a curve whose radius is 240 m. Each boat experiences the same centripetal acceleration. What is the ratio vA/vB of the speeds of the boats?
My physics optional extra work: This would be helpful so I can compare the solutions here as I do the work by myself.
Why did the speedboats go to therapy?
Because they were feeling a bit...curved out!
Now, let's solve this problem together. The centripetal acceleration, ac, is given by the equation ac = (v^2)/r, where v is the speed of the object and r is the radius of the curve.
For speedboat A, acA = (vA^2)/115.
For speedboat B, acB = (vB^2)/240.
Since both boats experience the same centripetal acceleration, we can set the two equations equal to each other:
(vA^2)/115 = (vB^2)/240.
Cross-multiplying the equation gives us:
240vA^2 = 115vB^2.
Finally, dividing both sides by 240vB^2, we get:
vA^2/vB^2 = 115/240.
Taking the square root of both sides, we find:
vA/vB = √(115/240).
So, the ratio of the speeds of the two boats, vA/vB, is √(115/240).
Now, back to therapy for those speedboats! They have a lot of curve-related issues to work through.
To compare the speeds of the two speedboats, we can use the formula for centripetal acceleration:
ac = v^2 / r
Where:
ac is the centripetal acceleration,
v is the velocity,
and r is the radius.
Given that both speedboats experience the same centripetal acceleration, we can equate their formulas:
(1) for speedboat A: acA = vA^2 / rA
(2) for speedboat B: acB = vB^2 / rB
Since the centripetal accelerations are equal, we have:
vA^2 / rA = vB^2 / rB
To find the ratio vA/vB of the speeds of the boats, let's solve for vA/vB:
vA^2 / rA = vB^2 / rB
Cross-multiplying:
vA^2 * rB = vB^2 * rA
Dividing both sides of the equation by vB^2 * rB:
vA^2 / vB^2 = rA / rB
Taking the square root of both sides:
vA / vB = √(rA / rB)
Substituting the given values:
vA / vB = √(115 / 240)
Now, we can calculate the ratio of the speeds of the boats.
To find the ratio of the speeds of boats A and B, we can use the formula for centripetal acceleration:
ac = (v^2) / r
where ac is the centripetal acceleration, v is the speed, and r is the radius of the curve.
Since we are given that both boats experience the same centripetal acceleration, we can set up the following equation:
(ac)A = (ac)B
Equating the two equations for centripetal acceleration and rearranging, we get:
(vA^2) / rA = (vB^2) / rB
Cross multiplying, we have:
vA^2 * rB = vB^2 * rA
Dividing both sides by vB^2 * rB, we get:
vA^2 / vB^2 = rA / rB
Taking the square root of both sides, we have:
vA / vB = sqrt(rA / rB)
Now, we can substitute the given values for the radii:
vA / vB = sqrt(115m / 240m) = sqrt(0.4792)
Using a calculator, the square root of 0.4792 is approximately 0.6927.
Therefore, the ratio of the speeds of boats A to B is approximately 0.6927.