Two identical twins, Amy and Beatrice, are on top of two different slides, one on each slide. Both slides start 3 m above the ground and end 0.5 m above the ground. Amy's slide is a straight, 45 degree slide with height given by h=−x+3 (so it starts at x=0 with a height of three meters and ends at x=2.5 with a height of half a meter). Beatrice's slide is curved and the height is given by h=3/(1+x) (so it starts at x=0 and ends at x=5). Both slides are frictionless.

If both twins start from rest at the top of their slides, what is the ratio of Amy's speed to Beatrice's speed at the bottom of the slides? i.e., what is vAmy/vBeatrice?

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-Calvin
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To find the ratio of Amy's speed to Beatrice's speed at the bottom of the slides, we need to determine their respective velocities at the bottom using their given height functions.

Let's start with Amy's slide. The height function for Amy's slide is given by h = -x + 3. To find her velocity at the bottom, we need to determine the change in height over the change in x.

The change in height is given by h_final - h_initial = (0.5) - (3) = -2.5 meters.
The change in x is given by x_final - x_initial = 2.5 - 0 = 2.5 meters.

Using the formula for average velocity, v = delta x / delta t, we can rearrange it to solve for delta t:
delta t = delta x / v.

Since Amy starts from rest, her initial velocity, v_initial, is 0. Therefore, her final velocity, v_final, is equal to the average velocity:
v_final = delta x / delta t = -2.5 / 2.5 = -1 m/s (Note: The negative sign indicates that the velocity is downward).

Now let's move on to Beatrice's slide. The height function for Beatrice's slide is given by h = 3 / (1 + x). Similarly, we need to find the change in height and change in x.

The change in height is given by h_final - h_initial = (0.5) - (3) = -2.5 meters.
The change in x is given by x_final - x_initial = 5 - 0 = 5 meters.

Using the formula for average velocity, we can find Beatrice's final velocity:
v_final = delta x / delta t = -2.5 / 5 = -0.5 m/s (Note: Again, the negative sign indicates a downward velocity).

Finally, we can find the ratio of Amy's speed to Beatrice's speed at the bottom of the slides:
vAmy / vBeatrice = (-1 m/s) / (-0.5 m/s) = 2.

So, the ratio of Amy's speed to Beatrice's speed at the bottom of the slides is 2. Amy will be moving twice as fast as Beatrice.

Shame on you Keshav!!! Cheating on Brilliant!!! This site is meant to be a platform to practice your own skills, not to copy paste the questions and get free answers and then get incentives without effort. So either play fair and be honest or leave this site. People like you are shame to the Brilliant community. And to the others, please give the answer to this problem after Monday 10/6/2013, so that this cheat doesn't get the opportunity to cheat.

By the way do you even understand the problem fully?