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When something does work on an object it changes the total energy of the object.  In this case, the work done by the person converts to kinetic energy as the stone is launched.  This kinetic energy is then turned into gravitational potential energy when the stone is at the highest point of the peak.

Work done = Kinetic Energy when the stone leaves the hand = Gravitational Potential Energy at the peak

We can set the work done to the gravitational potential energy

Plug in our known values and solve.

Known

We need to convert our velocity to 

Unkonwn

The easiest way to solve this problem is through conservation of energy.  When the car is rolling along the road it has kinetic energy.  Once the spring brings it to a complete stop, the car has elastic potential energy. According to the law of conservation of energy, these two values should be equal to one another

We can now plug in our known values and solve for the missing variable

The equation for potential energy is . We are given the mass of the ball, the height of the table, and the acceleration of gravity in the question. The distance the ball travels is in the downward direction, making it negative.

Plug in the values, and solve for the potential energy.

The units for energy are Joules.

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Equation

Plug in the vales and solve for the stretch of the spring.

Work causes a change in the kinetic energy of an object.  In the example of a car stopping, the work done on the car causes the car to slow down to a stop, therefore changing the kinetic energy.

Work is also equal to the force times the displacement of the object. In this case, we are assuming that the force applied to stop the car does not change.

Since the car is coming to a stop the final velocity of the car is 0m/s.

Therefore when you double the velocity, that value is then squared.

Therefore your kinetic energy is increased by  times the original amount.

If your kinetic energy is  times greater, than with the same force being applied, the stopping distance will also increase by  times since they are directly related.

Hooke’s law states that spring constant is directly related to the force applied and the distance that the object is stretched.

We also know that the potential energy of a spring is related to the spring constant and the distance that the object is stretched

We can substitute our equation for Hooke’s law into the potential energy equation.

This simplifies to

This equation shows that there is a direct relationship between the force on the spring and the potential energy of the spring.  If the force is doubled, then the potential energy will likewise double.

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First let us consider the force acting on the spring. Since the spring is hanging vertically, the only force acting on the spring is the force of gravity from the mass that has been added to the spring.

We can substitute in our variables and find the gravitational force.

According to Hooke’s Law the spring constant of a spring is directly proportional to the force acting on it and inversely proportional to the amount of stretch.

We can use the provided spring constant and the force acting on the spring to determine the amount of stretch.

To solve for the kinetic energy, we will need to use the equation:

Before we can plug in our given values, we must convert the mass from grams to kilograms. Remember, the SI unit for mass is kilograms, so most calculations will require this conversion.

Now we can use this mass and the given velocity to solve for the kinetic energy.

Therefore when you double the velocity, that value is then squared.

Therefore your kinetic energy is increased by 4 times the original amount.