Question

suppose you have two cans, one with milk, and the other with refried beans. Both cans have the same size, shape, and mass. If you release both cans at the same time, on a downhill ramp, which would have the most energy at the bottom of the ramp? ignore friction and air resistance..

a.can with beans

b. can with milk

c. both have same energy

please explain your answer

Answer 1

Concepts and reason

The concepts used in this problem are the concept of potential energy and the concept of conversion of potential energy into kinetic energy.

First, use the concept of conservation of energy to determine the expression for the kinetic energy at the bottom of the ramp. Later on, use the concept of the total energy, calculate the total energy for both cases to compare the result.

Fundamentals

Potential energy:

It is a form of energy. The potential energy is the stored energy in the body due to the position of the body.

In the gravitational influence, the potential energy of a body can be expressed as,

$U = mgy$

Here, potential energy is $U$ , the mass of the object is $m$ , the elevation of the object is $y$ and acceleration due to gravity is $g$ .

Kinetic energy:

The expression of the kinetic energy of a body can be described as:

$K = \frac{1}{2}m{v^2}$

Here, $K$ is the kinetic energy, $m$ is the mass, and $v$ is the velocity.

Conversion energy law:

According to the law of energy conservation, to the total energy of a system is conserved if there is no externally incoming and outgoing flow of energy.

The diagram of the problem is shown as,

From the diagram, it can be seen that both the cans are at the same height at $y = H$ an instant of time. Therefore they possess the same potential energy. At the other instance, they started rolling down at the same instant of time. They rolled and reached the bottom of the ramp. The height of the final position for both cans is zero and the potential energy has a direct proportionality relation with the height. Therefore the total potential energy will become zero. The energy can never be created nor be destroyed hence, the potential energy converts into the kinetic energy.

If the friction and air drag are neglected then there will be no potential energy loss and hence, the total potential energy will convert into the kinetic energy.

Hence, applying energy conservation law the total potential energy will be converted into the kinetic energy.

At the bottom line, the kinetic energy of the can is,

$\begin{array}{c}\\K = U\\\\ = mgH\\\end{array}$

When the milk can reach the bottom line, its total potential energy can be calculated as:

$U = mgy$

Substitute the value $0$ for $y$ .

$\begin{array}{c}\\U = mg \times 0\\\\ = 0\\\end{array}$

At the bottom line, the potential energy of the milk can is zero.

The kinetic energy of the can is,

$K = mgH$

At the bottom line, the kinetic energy of the milk can is $mgH$ .

The total energy of the milk can at the bottom line is.

${E_{\rm{m}}} = K + U$

Here ${E_{\rm{m}}}$ is the total energy of the milk can at the bottom line.

Substitute the value $0$ for $U$ and $mgH$ for $K$ .

$\begin{array}{c}\\{E_{\rm{m}}} = mgH + 0\\\\ = mgH\\\end{array}$

When the refried beans reach the bottom line, its total potential energy can be calculated as:

$U = mgy$

Substitute the value $0$ for $y$ .

$\begin{array}{c}\\U = mg \times 0\\\\ = 0\\\end{array}$

At the bottom line, the potential energy of the refried beans can is zero.

The kinetic energy of the can is,

$K = mgH$

At the bottom line, the kinetic energy of the refried beans can is $mgH$ .

The total energy of the refried beans can at the bottom line is.

${E_{\rm{r}}} = K + U$

Here ${E_r}$ is the total energy of the refried beans can at the bottom line.

Substitute the value $0$ for $U$ and $mgH$ for $K$ .

$\begin{array}{c}\\{E_{\rm{r}}} = mgH + 0\\\\ = mgH\\\end{array}$

Therefore it can be concluded that,

${E_{\rm{m}}} = {E_{\rm{r}}}$

Hence the total energy at the bottom of the ramp is the same for milk can and the refried beans can.

Ans:

The total energy at the bottom of the ramp is the same for milk can and the refried beans can.