Review Three cars are starting on a 30-mile trip. They start at the same time, and arrive ½ hour...
22
Review Three cars are starting on a 30- mile trip. They start at the same time, and arrive ½ hour later. Slow start, then becoming faster Constant speed Fast start, then becoming slower Which car has the highest average speed? Draw the x-t graph for the three cars. Draw the v-t graph for the three cars.
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Transcript of Review Three cars are starting on a 30-mile trip. They start at the same time, and arrive ½ hour...
Review
Three cars are starting on a 30-mile trip. They start at the same time, and arrive ½ hour later.
Slow start, then becoming faster
Constant speed
Fast start, then becoming slower
Which car has the highest average speed?
Draw the x-t graph for the three cars.
Draw the v-t graph for the three cars.
Reading assignment 2.2
Which of the following statements about the instantaneous velocity is false?
a) The inst. Velocity is equal to the slope of the tangent in an x-t graph.
b) The instantaneous velocity is equal to the speed at any time.
c) The instantaneous velocity is the derivative of position with respect to time.
d) The inst. Velocity can be written as x
dxv
dt
Reading Assignment 2.3:
The slope of which of these green lines could be used to find the average acceleration of a particle?
A. B.
C. D.
vx
t
vx
t
x
t
vx
t
Reading Assignment:
Which of the following is a correct equation for the instantaneous acceleration in x direction?
A. B. x
x
dva
dx
C. D.
x
dxa
dt
xx
dva
dt
2
2x
d xa
dv
Instantaneous velocity
Constant speed:• straight line in x-t graph• Slope = velocity = average velocity
What happens
if the velocity is
not constant?
Instantaneous velocity
x
t
The sled becomes faster as it goes downhill. Which of the following curves does best represent position versus time
for the sled?
A
B D
C
Instantaneous velocity
x
t
The sled becomes faster as it goes downhill. Which of the following curves does best represent position versus time
for the sled?
C
Instantaneous velocity
x
t
x(t) position x is a function of time t
t1
x1
t2
x2
x
t
ave
xv
t
velocity at time t1 ?
Instantaneous velocity
x
t
t1
x1
t3
x3
x
t
ave
xv
t
t2
velocity at time t1 ?
Instantaneous velocity
x
t
t1
x1
t3
x4 xt
ave
xv
t
t2t4
velocity at time t1 ?
Instantaneous velocity
x
t
t1
0t
ave
xv
t
velocity at time t1 ?
10
limt
xv
t
dx
dt
Instantaneous velocity
s
t
t1
velocity at time t1 ?
10
limt
xv
t
dx
dt
t1
Instantaneous velocity
• Instantaneous velocity
– is the first derivative of distance with respect to time
– Slope in x-t graph
• Unit of measurement: 1 m/s
• Speed is the magnitude of the instantaneous velocity
Instantaneous velocity - example
A robot is logging its position as a function of time, given below. Sketch the graph of velocity versus t.
Position x
Time t
t1t2
A ball is rolling up an inclined straight length of track. Its distance from the bottom of the track is described by the equation
a) Find an expression for the velocity as a function of time.b) Where was the ball when the clock read zero?c) Where is the highest x-position the ball can reach?d) Find the acceleration of the ball as a function of time.e) At which time does the ball fall off the bottom end of the track?f) Sketch the x-t graph, the v-t graph, and the a-t graph.
22
4.0 2.0 2.0m m
x m t ts s
You may have noticed while driving your car’s velocity does not continue to increase, even though you keep your foot on the gas pedal (drag, friction, engine efficiency). Shown is a v-t graph of a typical car starting from rest.
v
t
Sketch a-t and x-t graphs.
Example: 1-d acceleration
a) A school bus is approaching a stop and
slowing down from 15 mph to zero. This takes
4.5 seconds. Determine the average
acceleration of the school bus.
b) After the stop the school bus resumes its trip
by reaching 15 mph after 6 seconds. Find the
average acceleration.
Can displacement and velocity have opposite signs for a time interval?
a)Yes, in case of an accelerated motion.
b)No, since the sign of the displacement determines the sign of the velocity.
c)There is no relationship between the two quantities.
Can velocity and acceleration have opposite signs for a time interval?
a)Yes, if the speed is decreasing.
b)Yes, if the object moves to the left.
c)No, since the sign of the velocity determines the sign of the acceleration.
Motion in positive direction
Motion in negative direction
The sign of acceleration?fx ix
xf i
v va
t t
x
Increasing speed fxv
ixv xv
0xa
Decreasing speed
ixv
fxv xv
0xa
fxvixvxv
0xa
ixv
fxvxv
0xa
Sign of acceleration?
• Depends on– Coordinate system (choice)– Direction of motion relative to coordinate
system (choice)• Speed increases
– If velocity and acceleration have the same direction
• Speed decreases– If velocity and acceleration oppose each other
A skier is sliding downhill, starting from rest, at a constant acceleration of 0.5 m/s2.
a)What is his speed as a function of time?
b)After 5.0 s, how far did the skier slide, and how fast is he going?
