You climb a high mountain and lie down, exhausted, at the top. A hawk which started at the bottom floats lazily by and perches beside you, having risen to the mountain top on a thermal. Did it take more work to get you to the top of the mountain than it did to get the hawk there? a) Yes, you got to the top first, which requires more work. b) No, it takes the same amount of work to raise any two objects a height h. c) Yes, the rising thermal provides free energy which does not count as work. d) No, the hawk traveled a longer path. In physics terms it actually took more work to get the hawk to the top. e)Yes, because your mass is greater than the hawk’s.

Correct Answer – E a)It doesn’t matter who got there first or faster. Velocity and time do not affect the amount of work done, as long as you and the Hawk have the same velocity at the start and at the end. c) While its true that the thermal energy didn’t come from the hawk, it still counts as work done. The rising currents of hot air lost some of their energy lifting the hawk. d) In cases involving conservative forces, the length of the path does not matter, only where you begin and end. If we took account of friction or air resistance, the length of the path would matter. b) and e) It is true that the Work done in both cases is m g h, and h is the same for you and the Hawk, but clearly your mass is greater than the Hawk’s. So e) is the right answer.

When you push a block across a frictionless table you accelerate it, which means that the work you did is transformed into kinetic energy. Suppose the table was rough and the friction force balanced your pushing force so that the block started and finished with the same speed. You still did work, so where did the energy go? a)Sometimes energy just disappears. b)It is just like the case of raising the block, you’ve stored the energy somewhere and will be able to use it later in moving the block back. c)The energy turns into heat in the block and the table. d)The energy is stored in your muscles.

Correct Answer – C Friction is an example of a non-conservative force. A non- conservative force turns mechanical energy (meaning either kinetic energy or potential energy) into other forms of energy, like heat. The energy does not disappear, but we do lose all or some of our ability to do work with it, so the energy does lose much of its usefulness.

Kinetic Energy – Energy of Motion Suppose you push a block across an ice sheet with almost no friction. If you push with a constant force F for a distance d, what amount of kinetic energy, K, will the block have at the end of the push, assuming it was not moving at all beforehand? a)K = F b)K = F d c)K = F / d d)K = d e)K = d / F

Correct Answer – B The kinetic energy of the block after you finish pushing it will be equal to the work you did, if there are no non-conservative forces like friction involved. The Work you did was W = F d, so therefore the kinetic energy you gave the block is K = W = F d. You lost the same amount of energy (positive work means you lose energy, and the system you did work on gains the energy).

Now we want to calculate K, the kinetic energy of the block, in terms of the mass of the block, m, and the speed of the block, v. How fast will the block be going at the end of your push? Our force was constant, so it produced a constant acceleration a = F/m. We don’t know (and don’t care) how much time we pushed for, but we do know what distance the acceleration took place for. We also know the initial speed was 0. Which of the following is true? a)v = (F/m) t b) v 2 = ½ (F/d) m c) v 2 = 2 (F/m) d d) v = ½ g (F/m) 2

Correct Answer – C Given the information we had, it made sense to make use of the equation v 2 = v o a x = (F/m) d = 2 (F/m) d Let’s rearrange this a little to get ½ m v 2 = F d But we know that F d was the work we did on the block, which is equal to the kinetic energy we gave to it, so K = W = F d = ½ m v 2 Note that the fact that v 2 appears in this equation means that it is the speed, not the velocity of the block that counts, since you have the same kinetic energy whether you are driving north, south, east or west.

So we have our two main forms of mechanical energy, potential and kinetic. Suppose I take a stone and throw it through the air with horizontal velocity v x and vertical velocity v y. At the highest point of its flight, when its height above the ground is h and it is about to start coming down, what is its total mechanical energy? Assume that there is no air resistance, and that therefore total mechanical energy is conserved. a)K = m g h + ½ m v x 2 b)K = m g h c)K = ½ m v x 2 d)K = m g h + ½ m v y 2

Correct Answer – A We already know from our study of projectile motion that at the highest point of the flight the vertical velocity has been reduced to zero. Therefore this cannot contribute to the kinetic energy. But in the absence of air resistance the horizontal component of velocity is unchanged. Therefore all of the initial kinetic energy associated with the vertical part of the motion has been converted to potential energy.

Now let’s use conservation of mechanical energy to figure out how the height reached by the stone depends on the initial vertical velocity with which we threw the stone. a)h = v y 2 /(2 m g) b)h = m g v y 2 c)h = v y / (2 m) d)h = v y 2 / (2 g)

Correct answer – D Conservation of energy says that the initial energy and the final energy must be the same. Initially the total energy was E i = ½ m v x 2 + ½ m v y 2 The final energy was E f = ½ m v x 2 + m g h If we demand that E i = E f Then it follows that ½ m v y 2 = m g h which means that h = v y 2 / (2 g) Which is the same formula we previously derived using our kinematics equations.

When James Joule tried to take the temperature of the water at the top and bottom of his honeymoon waterfall, what amount of energy was he looking for? a)m g h, where m is the mass of water that falls over the waterfall, and h is the height of the waterfall b)½ m v 2, where m is the mass of water that falls over the waterfall, and v is the speed of the water as it flows horizontally along the river c)F d, where F is the force of the water as it strikes the rocks at the bottom of the waterfall, and d is the width of the turbulent pool at the bottom d)W t, where W is the patience of his wife and t is the time remaining before the first row of their marriage.

Correct Answer – A Answer A is the potential energy in the water at the top of the waterfall and answer B is its kinetic energy. Presumably the water will still flow when it reaches the bottom, so let’s assume that the kinetic energy is conserved. Given that, what Joule wanted to find out is what happened to the potential energy stored in the water before it fell. Did it turn into heat. Of course he found that the water was cooler at the bottom of the waterfall than at the top, which is something that most people who’ve been to a waterfall know (even if they don’t experience the decidedly chilly reception awaiting them after leaving their wife alone at the bottom for a length period on her honeymoon). Can anyone think where the missing energy has gone? Think of Joule’s wife sitting at the bottom with time on her hands. What atmospheric effect might she notice?

Joule’s wife might notice cool mist at the bottom of a tall waterfall. As the water falls it tends to form water droplets in the air. The total surface area of the water is greatly increased while it is falling. This means more of the water can turn into water vapor. To do this requires energy (latent heat of evaporation). So the potential energy is converted into latent heat. Even some of the water’s own heat energy is lost to this, cooling the water as it falls.