I'm doing my homework for a Physics class and trying to get ready for a quiz. I encountered this question:
Two ramps of equal length are situated such that ramp #1 has a slope (with respect to the horizontal) of 30°, and ramp #2 has a slope of 60°. Neglecting friction, roughly by what factor is the time it takes a ball to roll down ramp #1 larger than the time it takes a ball to roll down ramp #2?
I did this:
$$ \frac{\sin60}{\sin30} $$
That came out to be 1.74, but the correct answer is 1.3. How?
Best Answer
Note:
Not heights, but lengths. Your solution would be correct if the question was about equal heights.
The component of the gravity force in the direction of sliding is proportional to the sinus of the slope angle. Hence - the acceleration is also proportional to the sinus of the slope angle.
Now, given the constantly accelerated motion, the time it takes to pass a given distance is reverse proportional to the square root of the acceleration. That is:
X = a * t^2/2. Hence T = (2X/a) ^ (1/2).
T = (2X/g / sin(α) ) ^ (1/2).
T1 : T2 = [ sin(α2) / sin(α1) ] ^ (1/2)
If the question was about equal heights then balls would pass different distances (in the sliding direction). The distance would be divided by another sin(α) factor. Hence the answer would be
T1 : T2 = sin(α2) / sin(α1)