# How do I solve for total energy using kinetic and potential energy equations?

In summary, the distance a child travels is 8 meters when sin(30 degrees) is 4 divided by x. The conservation of energy states that the total energy of the system must be the sum of kinetic and potential energy. The child's potential energy is the sum of the kinetic energy and the distance they travel, while their kinetic energy is the product of their mass and the speed they are moving.

## Homework Equations

Total Energy = Kinetic Energy + Potential Energy
T.E = 1/2mv^2 + m(g)(h)

## The Attempt at a Solution

Distance Child travels
Sin 30 = 4/ x
x = 8m

T.E at top
= 1/2mv^2 + m(9.8)(h)
= 40m

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Conservation of energy only works for non-conservative forces! You could do E = Work_net = Work_conservative + Work_non-conservative, but going back to Newton's 2nd law should give an easier time.

well you're on the right track, and assume from your last calculation that you're rounding g to 10m/s^2.

TE at bottom=pe+ke where pe=0

however, there is energy lost in going from top to bottom in form of friction.

so TE at top=TE at bottom plus frictional energy. You are given a magnitude for friction and have computed the distance right, can you finish from here?

edit: this was more or less simultaneous post, I think its actually easier using energy eqn, but solveable from either approach.

40m = 1/2mv^2 + 1/4m
v = 12.6 - m

is this correct?

And I don't know how to calculate the mass.

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closer. this is how I approached the problem, and always BTW much better to complete the algebra before posting numbers--both for purposes here and on an exam as a simple number mistake made early will cost dearly;

mgh=1/2mv^2+Ff*distance where Ff=frictional force

we know from problem, that frictional force = 1/4mg

so mgh-1/4*(mg)*8m= 1/2mv^2
(8m from your calculations involving length of slide)

This is what I got:
40m-1/4*(m)*8m= 1/2mv^2
40m - 2m = 1/2mv^2
76m / m = v^2
v = 8.7 m/s

Question stated that friction is 1/4m not 1/4mg. Or is it suppose to be 1/4mg?

The question said the frictional force was one quarter of the child's weight. Weight is m*g, expressed in Newtons. So it should be (1/4)mg.

This is what I got now:
40m-1/4*(m)*8m= 1/2mv^2
40m - 20m = 1/2mv^2
20m / m = v^2
v = 4.47 m/s

Where did the 1/2 from the kinetic energy go? You dropped it in the second last line.

O you thanks I forget to multiply by 2.

This is what I got now:
40m-1/4*(m*g)*8m= 1/2mv^2
40m - 20m = 1/2mv^2
40m / m = v^2
v = 6.3 m/s

## What is the law of conservation of energy?

The law of conservation of energy states that energy cannot be created or destroyed, but can only be transformed from one form to another. This means that the total amount of energy in a closed system remains constant over time.

## How does the conservation of energy apply to everyday life?

The conservation of energy applies to everyday life in various ways, such as when we turn on a light switch, the electrical energy is converted into light and heat energy. It also applies to activities like driving a car, where the chemical energy in gasoline is converted into kinetic energy to move the car.

## What are some examples of energy transformation?

Energy transformation occurs in many forms, such as when a battery-powered device converts chemical energy into electrical energy. Other examples include burning wood, which converts chemical energy into heat and light energy, and photosynthesis, which converts solar energy into chemical energy in plants.

## How is the conservation of energy related to the first law of thermodynamics?

The conservation of energy is directly related to the first law of thermodynamics, which states that energy cannot be created or destroyed. This law is a fundamental principle of physics and is the basis for the law of conservation of energy.

## What are some real-world applications of the conservation of energy?

The conservation of energy has many real-world applications, including electricity generation, transportation, and environmental conservation. It is also used in industries such as manufacturing and construction to optimize energy usage and reduce waste.