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Workshop Tutorials for Technological and Applied Physics
Solutions to MR6T: Conservation of Energy
A. Qualitative Questions:
PE = mgh
KE = 0
1. a. and b. With no air resistance:
v=0
The ball has an initial velocity, vi, and kinetic energy KEi =½
h
2
mvi . It also has an initial height hi =0 and potential energy
PE = 0
v = vi
PEi = mghi = 0. As the ball goes up, its kinetic energy
KE = ½
decreases and is zero at a height of h, its gravitational
mvi2
potential energy increases and is maximum at h. The reverse
v = vi
PE = 0
happens on the way down, such that the total energy of the
2
KE
=
½
mv
i
ball is constant, i.e. at every instant PE + KE = total energy.
Or, in terms of work rather than potential energy, there is no change in kinetic energy of the ball
between the initial and final positions thus the total work is zero. During the flight the work done by
weight on the way up is W=-mgh and on the way down is W = mgh.
a. and b. With air resistance:
PE = mgh’ < ½ mvi2
As the ball goes up, there is work done by air resistance,
h’
KE = 0
so the ball’s its kinetic energy decreases and is zero at a
v=0
height h' which is less than h. On the way down again
there is work done by air resistance. Hence the final
PE = 0
v = vi
kinetic energy is less than the initial kinetic energy, so
KE < ½ mvi2
final speed is less than initial speed. The total energy of
the ball-earth system is PE + KE + Wair resistance.
v < vi
PE = 0
Total energy is constant but the ball has less mechanical
2
KE
=
½
mv
i
energy when caught. Some of its energy has been
converted into heat due to work done by air resistance.
c. The energy is converted into heat, sound and motion of the muscles in your hand.
d. Your hand must do work on the ball to change its kinetic energy from ½ mv2 to 0. The work done is
given by the force times the distance, so if you increase the distance over which your hand applies the
force to stop the ball, the force required is less. If the force on the ball by your hand is less then the force
by the ball on your hand will also be less.
2. Gravitational potential energy.
a. The height h in gravitational potential energy, mgh, depends on the situation being analysed. It is often
measured from the surface of the Earth. This is an arbitrary but convenient zero point, as all we can really
measure is changes in energy anyway.
b. If you define the surface of the Earth as the zero gravitational potential energy, then anywhere below
the surface is negative, for example down a mine or well.
c. It is not possible to have a negative kinetic energy as this would imply a negative mass. (Even if the
velocity is negative the KE is ½ mv2, so is still positive.)
d. We can’t define an absolute value for either gravitational or kinetic energy. For gravitational potential
energy we define it relative to some height, and for kinetic energy we define the movement as being
relative to something. Your velocity relative to the Earth’s surface is different to your velocity relative to
a moving train, so depending on your frame of reference you may have different kinetic energies.
B. Activity Questions:
1. Bouncing balls I
Balls that lose less energy to non-mechanical forms rise higher than balls that lose more energy. A ball
can bounce higher than the original height if we throw the balls instead of just dropping them. These balls
start with kinetic and potential energy instead of just potential energy
The Workshop Tutorial Project – Solutions to MR6T: Conservation of Energy
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2. Solar panels and electric circuit
a. Energy as light is converted to electrical energy by the
solar cell which is then converted to kinetic energy by the
motor. Some energy is also converted to heat, which is
usually not considered useful.
b. Heat is useful energy when you want heating or cooking.
c. Dammed water has gravitational potential energy, which
is converted to kinetic energy when the dam is open. This is
converted to kinetic energy of a turbine placed in the flow,
which is attached to a generator. The generator turns the
kinetic energy into electrical energy which is converted into
light, heat sound or mechanical energy by a home appliance.
lamp- electrical energy to
light energy and heat
solar cell- light energy
to electrical energy and
heat
3. Tennis racquet
A tennis racquet that is held in your hands vibrates. Some of the
kinetic energy of the ball on impact is transferred to the racquet and
some is transferred to your wrist, hands and arms thus the ball has a
smaller kinetic energy when it bounces up and rises to a lower height.
A racquet that you step on cannot vibrate and thus energy is not
transferred to your ankles and foot. Thus the kinetic energy of the ball
just before impact is approximately equal to the kinetic energy of the
ball just after impact and it rises to almost the height it was dropped
from.
C. Quantitative Question:
motor- electrical energy to
kinetic energy and heat
lots of
energy
lost
vibrates
fixed
little
energy
lost
friction
f= N
= mgcos30o
1. Brent on the banister.
a. See diagrams opposite
N=mgcos30o
b. Frictional force and the gravitational force do work on Brent.
c. Initially Brent is at rest and has potential energy but no kinetic
mgsin30o
mgcos30o
energy. As he slides the potential energy is converted to kinetic energy
and thermal energy.
d. Initial PE = final KE + thermal energy.
vo =0
e. Initial PE = mgh = mgLsin30o; final KE = ½ mv2,
thermal energy produced = fL = mgcos30oL
3m
and using d : mgLsin30o = ½ mv2 + mgcos30o
30o
we get v 2 = 2gL (sin30o - cos30o) = 3.9. Therefore v = 2.0 m.s-1.
f. The average rate at which mechanical energy is dissipated is the average power which is <P> = F<v>
and F = f = mgcos30o, where <v> = ½ (vo+v) = 0.99 m.s-1, so <P> = mgcos30o<v> = 0.29 kW.
2. Coyote on a spring.
a. The spring constant is: k = F/x = mg/x = (16 kg  9.8 m.s-2 )/ 0.1 m = 1.6 kN.m-1.
b. When the spring is compressed by a further 50 cm, the total compression is through 60 cm or 0.60 m.
The elastic potential energy is U = ½ kx2 = ½  1600 N.m-1  (0.60 m)2 = 0.28 kJ
c.
U = 0.28 kJ
d. The coyote rises by a distance y. So ½ kx2 = mgy which we rearrange to give
kx 2
282J
y = 2 mg = 16 kg9.8 m.s  2 =1.8 m. This is not even close to the 10 m height he needs.
e. The energy of the oscillating spring comes from the kinetic energy of the spring just as the coyote
loses contact.
f. The total energy remains constant; energy is being converted from one form to another; kinetic energy
of both coyote and spring, gravitational potential energy of the coyote and the spring potential energy.
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The Workshop Tutorial Project – Solutions to MR6T: Conservation of Energy