2b. motion in one dimension
TRANSCRIPT
Position functions
A function which calculates the position of something
using:
• Displacement
• Velocity
• Acceleration
• Time
Position Functions
A function of x
A function of time
Distance
Distance as a function
of time
f(x)
f(t)
d
d(t)
Graphs
Sketch the following position functions using displacement, velocity and acceleration
1. A tree standing 100m from the origin (a house)2. A light rail train between stations3. An apple falling from a tree
Position functions
d(t) =
d(t) =
d(t) =
How can we calculate the position of a body with constant:
1.Displacement?
2.Velocity?
3.Acceleration?
Rates of change
• Note that:–Displacement’s rate of change is
measured by …• Velocity (d/t=v) [d’(t)]
–Velocity’s rate of change is measured by …• Acceleration (v/t=a) [d’’(t)]
Back to position functions
d(t) = c
d’(t) = v
d’’(t) = a
d(t) = [d’(t)].dt = v.dt = vt + c
v(t) = [v’(t)].dt = a.dt = at + c
d(t) = v.dt = (at + c).dt = 1
/2at2
+ ct + e(what do ‘c’ and ‘d’ represent?)(‘d’ think of a light rail between stations)(‘c’ why don’t supersonic jets shoot themselves down?)
In summary
Generally
d(t) = c
= vt + c
= 1
/2at2
+ vt + c
v(t) =at + v0
When x0=0 and v0=0
d(t) = 0 = vt
= 1/2at2
Simple harmonic motion
Sketch the position, velocity and
acceleration functions for a simple
harmonic oscillator, (assume no
energy is lost)
d(t) = cos(t)
v(t) = -sin(t)
a(t) = -cos(t)
One dimensional motion with constant acceleration
The Earth’s gravitational Force becomes weaker as you get further from its surface
However, at the surface, in the absence of air resistance,
all things fall with an acceleration of 9.8ms-2
(g).
Under these conditions, a feather and a piano would accelerate at the same rate.
Too warm Too cold Scared
PufferFalls
asleepSwims fast in circles
Blows up into a ball
BlueyPresses
against the cool glass
Swims into the plants
Swims in random
directions
FlattyJumps out
of the water
Tries to swim into Puffer's mouth
faints
• The plants are moving in a strange way. What is Puffer doing?• Puffer is getting annoyed at Flatty. What is Bluey doing?• Flatty is having trouble breathing. Why can't Puffer help her?• The fish find that they have very little room to swim. What is
Flatty doing?•
Interpret what the first
fish has done
Find out what situation
caused this
Find out what the
second fish does in this
situation
Interpret which value
should go into which
Question 1
How long would it take for an object, dropped from a height (h) to reach the ground?
d(t)= -
1/2at
2 + v0t + d0
0 = -1
/2gt2
+ 0.t + h1
/2gt2
= h
t2
= 2h
/g
t
=+
/-(2h
/g)1/2
Question 2
A bullet is fired directly upwards at an initial
speed of v0. How fast will the bullet be travelling
when it returns and hits the Earth?
d(t) = -1
/2at2
+ v0t + d0
0 = -1
/2gt2
+ v0t + 0
0 = t(v0-1
/2gt)
t = 0 or v0 -1
/2gt = 0
v0 = 1
/2gt
t = 2v0/g
v(t) =
Substituting 2v0/g for t
v(t) = -g(2v0/g) + v0
= -v0
ie: the same initial speed in the opposite direction
x’(t) = -gt + v0
Question 3
A bouncy ball has the property that if it hits the
ground with velocity ‘v’, it bounces back up with
velocity -0.8v. If this ball is dropped from a height
‘h’ above the ground, how high will it bounce?
1. Position function in freefall:
• d1(t)= -1/2gt2 + h2. Position function after bouncing:
• d2(t)= -1/2gt2 + vbt
Equation 1: x1(t)= -1/2gt2 + h
Time to hit the ground
0 = -1/2gt
2 + h
t = (2h
/g)1/2
Velocity when hitting the ground
v(t) = -gt + v0
v(t) = -g(2h
/g)1/2
= -(2ggh
/g)1/2
= -(2gh)
1/2
Hence, vb= +0.8(2gh)1/2
(in the opposite direction)
Equation 2: x2(t)= -1/2gt2 + vbt
Velocity function:
v(t) = -gt + vb
Velocity at maximum height:
0 = -gt + vb
t = vb/g
Substituting back into the position function:
d2(t)= -1
/2gt2
+ vbt
= -1
/2g(vb/g )2
+ vb(vb/g)
= -(vb2
g /2g2
) + (vb2
/g)
= -(vb2
g /2g2
) + (vb2
/g)
= -vb2
g + vb2
2g
2 g
= -vb2
+ 2vb2
2g
2g
= vb2
2g
Substituting:
0.8(2gh)1/2
for vb,
vb2
= 0.82(2gh)
2g 2g
= 0.64h