USC2001 Energy Lecture 1 Potential and Kinetic Energy

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USC2001 Energy Lecture 1Potential and Kinetic
Energy

• Wayne M. Lawton
• Department of Mathematics
• National University of Singapore
• 2 Science Drive 2
• Singapore 117543

Email matwml_at_nus.edu.sg Tel [phone]
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HEIGHT
Why is height a relative concept ?
How does it depend on a reference position ?
How does it depend on a reference length ?
Does the distance that an object fall depend on
the reference position ? Does it depend of the
reference length ? Does A or B report a longer
distance dropped ?
As reference length one stick
Bs reference length one stick
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LENGTH CONVERSION
Question If As stick has length c times the
length of Bs stick, how to convert from length
in A-sticks into length in B-sticks ?
Consider the equation length L x (A-sticks) y
(B-sticks)
Substitute the equation 1 (A-sticks) c
(B-sticks) to obtain x c (B-sticks) y
(B-sticks)
Question How is y related to x ? Why ?
Question How is x related to y ? Why ?
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HEIGHT CONVERSION
Question How are As and Bs height measurements
related if they both use the same reference point
(ie a point that has height zero) ?
Question What happens if As reference point is
d (B-sticks) above Bs reference point ?
Question Let a point have height u (A-sticks) in
As world and v (B-sticks) in Bs world. Derive
an equation that expresses u in terms of v and
another equation that expresses v in terms of u
(both equations will also involve both c and d)
Question Graph u as a function of v and v as a
function of u. What is the geometry of c and d
?
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CLOCKS AND INVARIANCE
How can we compare two time durations that start
at different times?
Our duration-length analogy might provide a clue!
Clue Find a repeatable physical process whose
start and finish are points in time.
Examples Sandclock, Pendulum, Spring
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THE MOTION OF A FALLING OBJECT
Empirical measurements
show that an object in a vacuum and starting with
zero velocity falls a distance L L(T) gT2 /
2 in a time interval having length T. Here g is a
constant equal to 9.8 meters per squared second.
Question according to this formula does L depend
on the position in time when or the position in
space where the object starts to fall ? does it
depend of the mass of the object ?
Question express height H H(t) as a function
of time t given H(t_0) and graph it
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THE SUPERPOSITION PRINCIPLE
Empirical measurements
show that the distance that an object in a vacuum
drops satisfies the following superposition
principle the distance it drops if it starts
with a nonzero velocity (positive for upwards and
negative for downwards) is the sum of the
distance that it would fall if it moved with
constant velocity and the distance that it would
fall if its initial velocity were equal to zero
Question express height H H(t) as a function
of time t given H(t_0) and V(t_0) and graph it
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GRAPHICAL REPRESENTATION
Consider a particle thrown upward from the ground
We can use analytic geometry to construct a graph
of this function using orthonormal planar
coordinates
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AVERAGE AND INSTANTANEOUS VELOCITIES
The average velocity over the interval
is the slope of the dotted line
Tangent
Secant
The instantaneous velocity at is the
slope of the solid line and equals the derivative
dh/dt evaluated at
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COMPUTING INSTANTANEOUS VELOCITIES
The instantaneous velocity v(t) is computed
using differential calculus
If the graph
of v is shown above
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COMPUTING DISTANCE FROM VELOCITY
The fundamental theorem of calculus says that h
equals the signed area under the graph of v
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TUTORIAL 1
1. Two observers A and B measure the heights of a
point x and a point y to be H_A(x)
312, H_A(y) 512 H_B(x) 125, H_B(y) 290
Compute constants (real numbers) r and c such
that for any point z H_A(z)
r H_B(z) c
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TUTORIAL 1
2. Three clocks A, B, C run at different
(uniform) rates and have simultaneous readings,
expressed in seconds, shown by the figure below?
Express the readings of B and of C as functions
of the reading of A. Hint consider the previous
tutorial problem
312
512
A
25
125
200
290
B
92
142
C
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TUTORIAL 1
3. A gun is fired directly at an object thrown
directly upward when its height is maximum
height. Show that the bullet will strikes the
target unless it hits the ground before the
object hits the ground (ignore air friction and
assume that the ground is flat). Hint the
horizontal motion is superimposed on the vertical
motion of the bullet fired from the gun. Also
ignore the effects of wind and friction and
Earths rotation.
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TUTORIAL 1
4. Compute the velocity V(t) for a falling object
given H(t_0) and V(t_0). Show that the quantity M
V(t)2 / 2 M gH(t) does not change with time.
The first term is called kinetic energy and the
second is called (gravitational) potential energy.