Work energy theorem problems cracking my heard

[farai]

Hello members

Please could you help me with solutions for the following two problems that I am stuck with.

King Kong is capable of jumping to a maximum vertical height of 10 m. He picks up Enrico Fermi, who is exactly 10% of Kong's mass, and leaps upwards. To what maximum height can King Kong leap to while he is carrying Enrico?

A skier starts at rest on the top of a strange, smooth, icy hill shaped like a hemisphere. The hill has a constant radius of R. Neglecting friction (it is icy!), show that the skier will leave the surface of the hill and become air-borne at a vertical distance of h = R/3, measured from the top of the hill.

Many thanks.

Farai

 

[futb0l]

1. Think of the gravitational potential energy.
2. Draw a diagram, label points and show the energy at those points. Energy is conserved.

 

[wais]

Hi Farai,

I would be happy to help you with these problems. Let's start with the first one about King Kong and Enrico Fermi. The key concept to remember here is the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy. In this case, we can use this theorem to determine the maximum height that King Kong can leap to while carrying Enrico.

First, let's define some variables. Let H be the maximum height that King Kong can leap to while carrying Enrico, and let h be the maximum height that King Kong can leap to without carrying Enrico. We know that King Kong can jump to a maximum height of 10 m without carrying Enrico, so we can write this as:

h = 10 m

Now, let's consider the work done on King Kong when he jumps with Enrico. The work done on King Kong is equal to the change in his kinetic energy. Initially, King Kong and Enrico are at rest, so their kinetic energy is zero. When King Kong jumps, he gains kinetic energy, which we can calculate using the formula:

KE = 1/2 * m * v^2

where m is the mass of King Kong and v is his velocity. Since Enrico is 10% of King Kong's mass, we can write this as:

KE = 1/2 * (1.1m) * v^2

where m is the mass of King Kong. Now, we can use the work-energy theorem to equate this to the work done on King Kong, which is equal to the force of gravity (mg) multiplied by the distance he jumps (H-h). This can be written as:

1/2 * (1.1m) * v^2 = mg * (H-h)

Solving for H, we get:

H = h + v^2 / (2g)

Now, we know that King Kong can jump to a maximum height of 10 m without carrying Enrico, which means that h = 10 m. We also know that King Kong can jump to a maximum height of 10 m, so we can write this as:

v^2 = 2gh

Substituting this into the equation for H, we get:

H = 10 m + 2gh / (2g)

Simplifying, we get:

H = 10 m + h