Download CONSERVATION OF ENERGY

CONSERVATION OF ENERGY
Theory
The kinetic energy of an object is defined as
and its gravitational potential energy is
1
𝐾 = 𝑚𝑣 2
2
𝑈𝑔 = 𝑚𝑔ℎ
where m is the mass of the object, v is its speed, g = the acceleration due to gravity, and h is the height
measured with respect to a reference level (usually the ground) where we define the potential energy to
be zero.
The law of conservation of mechanical energy states that if only conservative forces such as gravity do
work on a system, then its total mechanical energy remains constant throughout its motion. For a simple
system where the only force present is gravity, the total mechanical energy at any time is the sum of the
kinetic energy and gravitational potential energy. When we compare the energy at two instants A and B
we find
𝐾𝐴 + 𝑈𝑔𝐴 = 𝐾𝐵 + 𝑈𝑔𝐵 .
I. Testing a Theory of Cratering
A simple theory of cratering states
that when a large projectile crashes into the Earth, the crater produced
has a diameter D that is related in a simple way to the kinetic energy of the projectile at the time of
impact. We will test this theory by dropping two different steel balls from different heights into sand and
measuring the diameters of the craters produced. We don’t have an easy way to measure the kinetic
energy of impact directly, so instead we’ll use the fact that mechanical energy is conserved as the ball
falls. If we release the ball from rest and define h = 0 to be the surface of the sand, then
�𝑈𝑔 �
𝑇𝑂𝑃
= 𝐾𝐼𝑀𝑃𝐴𝐶𝑇
𝐷 ∝ 𝐾 𝑋 = 𝑈 𝑋.
Procedure
1. Drop each of the two balls into the sand, starting with the balls at varying heights, measured from the
center of the ball to the smoothed surface of the sand. For each crater, measure the diameter D of the
crater along two different axes and calculate the average value D of the diameter. After each
measurement, use your ruler to smooth out the surface of the sand.
1
6.35 mm diameter steel ball (Mass M =1.05 g )
h (m)
U (J)
D1 (m)
D2 (m)
D (m)
D2 (m)
D (m)
0.10
0.20
0.30
0.50
0.70
1.20
1.40
2.00
12.7 mm diameter steel ball (Mass M = 8.40 g)
h (m)
U (J)
D1 (m)
0.10
0.20
0.30
0.40
0.70
1.00
1.80
2. Use Excel to graph D vs U combining the data for both balls. Use a power fit to determine the value
of X. Print a copy of your graph (or get your TA to approve it).
The simple theory of cratering states that X = 0.25.
Value of X from your graph:
Percentage error from the theoretical result:
3. Meteorite A has a mass that is twice as great as meteorite B. It impacts the desert with a speed that is
1/3 the speed of B. What is the ratio of the crater diameters formed by A and B?
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II. Ramp & Toy Car as a Projectile
Equipment: ramp, platform, toy car, meter stick
h
H
D
Place the ramp on a 30 cm high platform. Place the toy car near the top of the ramp, and release it from
rest, so that it rolls down the ramp and is launched as a projectile onto the table. The car should be
moving in the horizontal direction as it leaves the ramp. You will use the principle of conservation of
energy along with the equations of projectile motion to predict the horizontal distance D the car travels
through the air. In order to make your prediction, you will need to measure only the vertical heights h
and H, shown in the figure. But first you must derive the desired equation for D.
Use conservation of energy to derive an expression for the car’s speed v as it leaves the ramp in terms of
the car’s initial height h above the launch point. Show your work on the back of this sheet.
v=
The car moves through the air a horizontal distance D as it falls from a height H. Use the kinematic
equations for projectile motion to derive an equation for D in terms of H and v. Show your work.
D=
Combine your equations for v and D to obtain an expression for D in terms of h and H.
D=
Measure the heights h and H, and use the expression for D to compute a predicted theoretical value for
D. Then measure D and compare it with your theoretical value.
Height h (cm)
Height H (cm)
Predicted D (cm)
Measured D (cm)
Percent Error
Given that the energy of the car depends on its mass, why were you able to use the principle of
conservation of energy to predict the distance D without ever having to measure the mass of the car?
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