A 2.5 g ice flake is released from the edge of a hemispherical bowl whose radius r is 40.0 cm. The flake-bowl contact is frictionless.

(a) What is the speed of the flake when it reaches the bottom of the bowl?

(b) If a second flake with twice the mass was substituted, what would its speed be?

Should I start by using KE=(1/2)mv^2? I am not really sure how I should approach this.

please disregard this, I've figured it out

I have so far converted 2.5g to 0.0025kg and 40.0cm to 0.4m.

An energy method is a good way to get the speed at the bottom.

At the bottom, (1/2) M V^2 = M g R (the P.E. loss).
Note that mass M cancels out

V= sqrt (2 g R)

a and b have the same answer.

The TIME it takes to reach the bottom is a lot harder to calculate, and requires solving a messy differential equation.

Oh, I see you're dealing with some physics equations. Let me break it down for you in a fun and humorous way!

(a) To find the speed of the ice flake when it reaches the bottom of the bowl, you can definitely use the equation KE=(1/2)mv^2. This equation relates the kinetic energy (KE) of an object to its mass (m) and velocity (v).

Since the ice flake is released from the edge of the bowl, it will experience a gravitational force acting on it as it falls towards the bottom. This force will cause the flake to gain speed and therefore kinetic energy.

To calculate the final velocity, you need to consider the conservation of energy. The initial potential energy of the flake at the edge of the bowl will be converted into kinetic energy as it falls to the bottom of the bowl. So you can set the initial potential energy equal to the final kinetic energy.

But wait! Since the flake-bowl contact is frictionless, we don't have to worry about any additional factors affecting the energy transfer. That makes our calculations a bit easier, doesn't it?

So, to start, you can calculate the initial potential energy of the flake at the edge of the bowl using the formula PE=mgh. Since the flake is released from the edge, the height (h) will be the radius of the bowl (r).

After that, you can equate this potential energy with the final kinetic energy, giving you the equation PE=KE. With a little math magic, you can solve for the final velocity (v)!

(b) Now, let's spice things up by substituting a flake with twice the mass! That's like throwing a clown with extra balloons into the mix. The only change here is that you'll need to update the mass (m) in your equation.

So go ahead and double the mass in the equation KE=(1/2)mv^2 to find the new final velocity.

And remember, physics can be a bit slippery sometimes, so watch out for those icy curves along the way!

Yes, you are on the right track by using the kinetic energy formula, KE = (1/2)mv^2. To find the speed of the flake when it reaches the bottom of the bowl, you'll need to consider the potential energy and kinetic energy of the flake. Let's break it down step by step:

(a) To find the speed of the first flake when it reaches the bottom of the bowl, we need to equate the initial potential energy to the final kinetic energy.

The potential energy (PE) of an object at height h in a gravitational field is given by PE = mgh, where m is the mass, g is the acceleration due to gravity (usually 9.8 m/s^2), and h is the height.

Since the flake is released from the edge of the bowl, the initial height is equal to the radius of the bowl, r. Therefore, PE_initial = mgh = mgr.

The kinetic energy (KE) of an object moving at a speed v is given by KE = (1/2)mv^2.

To find the speed when the flake reaches the bottom of the bowl, we set the initial potential energy equal to the final kinetic energy:

mgr = (1/2)mv^2.

We can now cancel the mass, m, on both sides of the equation, giving:

gr = (1/2)v^2.

To solve for the speed, v, rearrange the equation:

v^2 = 2gr.

Now take the square root of both sides:

v = sqrt(2gr).

Since we are given the value of the radius, r = 40.0 cm = 0.40 m, and the acceleration due to gravity, g ≈ 9.8 m/s^2, we can substitute those values into the equation to calculate the speed.

(b) To find the speed of the second flake, which has twice the mass, you'll follow the same steps as before, but with the new mass.

So, for the second flake, the mass is 2 × 2.5 g = 5 g = 0.005 kg.

Therefore, you'll substitute the value 0.005 kg for the mass, m, in the equations above to find the speed of the second flake when it reaches the bottom of the bowl.

Remember to convert the answer to the appropriate units, as the mass was given in grams but should be used in kilograms for the equations and calculations.

I hope this explanation helps you understand how to approach and solve this problem!