physicsssss?

Question

A boulder is raised above the ground so that its potential energy relative to the ground is 250 J. Then it is dropped. What is its kinetic energy just before it hits the ground?

Suppose an automobile has 1100 J of kinetic energy. When it moves at twice the speed, what will be its kinetic energy?
What's its kinetic energy at three times the speed?

100 N block of ice a vertical distance of 1 m is is raised the same vertical distance by pushing it up a 1.67 m long inclined plane, only 60 N of force is required. Calculate the work done to push the block up the plane.
What PE does it have?

Answer 1

1)
According to the conservation of energy law, the total energy (sum of the kinetic and potential energy) is constant. Before the boulder is dropped, its kinetic energy is zero (because its at rest), and all of its energy is in the form of potential energy relative to ground. Right before the boulder hits the ground, its potential energy relative to the ground is zero, and all of its energy is in the form of kinetic energy. So its kinetic energy before hitting the ground must equal the potential energy before it was dropped:
K = 250J

2)
Kinetic energy is proportional to the SQUARE of the velocity (K=½mv²). So when the velocity is doubled, the kinetic energy is quadrupled. When the velocity is tripled, the kinetic energy is nine times the original kinetic energy.

3)
The work done is the applied force times the distance in the direction of the force. Here, the apple force is 60N, and the distance in the direction of this force is 1.67m. So the work done on the block:
W = Fd = (60)(1.67) = 100 J
Note:
This is the same amount of work done had you simply lifted the block a distance of 1m (which you did, in a sense). The reason they are equivalent is that you are only doing work against the force gravity, which acts in the vertical direction. There is no resistive force in the horizontal direction (ignoring friction).
The gravitational potential energy, relative to its original position is:
PE = mgh = (100N)(1m) = 100 J
By lifting the block, you increased its gravitational potential energy.

Answer 2

PE = KE = 250 J, all the potention energy is converted to kinetic energy just prior to impact.

KE = 1/2 mv^2; so KE' = KE (mv'^2/mv^2) = KE (v'/v)^2; where m = mass of the car, v = initial velocity of the car, v' = 2v = doubled velocity, so that KE' = KE (2)^2 = 4 KE: doubling velocity quadruples KE. Similarly, tripling v' = 3v would make KE' = 9 KE, nine times the original kinetic energy, where KE = 1100 J.

PE = mgh; where mg = 100 N and h = 2 meters = height above the floor/ground while on the ramp. It's 2 meters because you started at 1 meter above ground/floor and pushed it up another 1 meter.

Work = Fd = 60 N X 1.67 m = number of Joules; where d = 1.67 m along the ramp and F = 60 N along the ramp.

A couple lessons learned here: first, PE is measured relative to ground level, whatever that is. This is so because the conservation of energy says that when a body is released unincumbered from a height h > 0, all the PE = mgh = KE = 1/2 mv^2 upon impact with that ground level where h = 0.

Second, work depends solely on a force acting over a distance. And that force is the force that is pushing in whatever direction d is. Going up the ramp, the boulder's weight (W) is not acting in the direction up the ramp, it's acting vertically downward towards the ground. So the F = 60 N and the weight (W) are two forces acting on that boulder. Of which, only F determines the work done.

Answer 3

Using the law of conservation of mechanical energy (and provided that we don't take air resistance into consideration) then the Kinetic energy just before the body hits the ground, equals its initial potential energy, i.e. 250 J.

Since K=(1/2)*m*v^2, when the speed doubles we have to multiply energy by 4. So the result is 4400 J.

Since the work needed is irrelevant of the path of the force (this is what we call conservative force fields) the same work is needed. So the answer is 100 N * 1 m = 100 J, provided that it has zero velocity. When the block reaches the top point, its potential energy will be exactly 100 J.