Chapter 2 Motion Along a Straight Line
Jan 19, 2016
Chapter 2
Motion Along a Straight Line
Linear motion
In this chapter we will consider moving objects:
• Along a straight line
• With every portion of an object moving in the same direction and at the same rate (particle-like motion)
Types of physical quantities
• In physics, quantities can be divided into such general categories as scalars, vectors, matrices, etc.
• Scalars – physical quantities that can be described by their value (magnitude) only
• Vectors – physical quantities that can be described by their value and direction
Distance, position, and displacement
• Distance (scalar) a total length of the path traveled regardless of direction (SI unit: m)
• In each instance we choose an origin – a reference point, convenient for further calculations
• Position of an object (vector) is described by the shortest distance from the origin and direction relative to the origin
• Displacement (vector) – a change from position xi to position xf
if xxx
Velocity and speed
• Average speed (scalar) - a ratio of distance traveled (over a time interval) to that time interval (SI unit: m/s)
• Average velocity (vector) - a ratio of displacement (over a time interval) to that time interval
• Instantaneous velocity (vector) – velocity at a given instant
• Instantaneous speed (scalar) – a magnitude of an instantaneous velocity
t
xvavg
t
xv
t
0
lim
if
if
tt
xx
dt
dx
Velocity and speed
Velocity and speed
Instantaneous velocity
• The instantaneous velocity is the slope of the line tangent to the x vs. t curve
• This would be the green line
• The light blue lines show that as Δt gets smaller, they approach the green line
Acceleration
• Average acceleration (vector) - a ratio of change of velocity (over a time interval) to that time interval (SI unit = (m/s)/s = m/s2)
• Instantaneous acceleration (vector) – a rate of change of velocity at a given instant
2
2
dt
xd
t
vaavg
if
if
tt
vv
t
va
t
0
limdt
dv
dt
dx
dt
d
Acceleration
• The slope (green line) of the velocity-time graph is the acceleration
• The blue line is the average acceleration
Chapter 2Problem 15
An object moves along the x axis according to the equation x(t) = (3.00 t2 - 2.00 t + 3.00) m, where t is in seconds. Determine (a) the average speed between t = 2.00 s and t = 3.00 s, (b) the instantaneous speed at t = 2.00 sand at t = 3.00 s, (c) the average acceleration between t = 2.00 s and t = 3.00 s, and (d) the instantaneous acceleration at t = 2.00 s and t = 3.00 s.
Case of constant acceleration
• Average and instantaneous accelerations are the same
• Conventionally
• Then
0it
t
vaa avg
if
if
tt
vv
0
t
vv if
atvv if
tt f
Case of constant acceleration
• Average and instantaneous accelerations are the same
• Conventionally
• Then
0it tt f
t
xvavg
0
t
xx if tvxx avgif
2221 fi
avg
vvvvv
2
atvi
if
if
tt
xx
2
)( atvv ii
2
2attvxx iif
Case of constant acceleration
dt
dxv
dt
dva
atvv if
2
2attvxx iif
Case of constant acceleration
Case of constant acceleration
To help you solve problems
Equations Missing variables
2
2attvxx iif
atvv if
)(222ifif xxavv
2
)( tvvxx fiif
2
2attvxx fif
fv
if xx
t
a
iv
Chapter 2Problem 28
A particle moves along the x axis. Its position is given by the equation x = 2 + 3t - 4t2, with x in meters and t in seconds. Determine (a) its position when it changes direction and (b) its velocity when it returns to the position it had at t = 0.
Case of free-fall acceleration
• At sea level of Earth’s mid-latitudes all objects fall (in vacuum) with constant (downward) acceleration of
a = - g ≈ - 9.8 m/s2 ≈ - 32 ft/s2
• Conventionally, free fall is along a vertical (upward) y-axis
gtvv if
2
2gttvyy iif
Chapter 2Problem 38
A ball is thrown directly downward, with an initial speed of 8.00 m/s, from a height of 30.0 m. After what time interval does the ball strike the ground?
Alternative derivation
Using definitions and initial conditions
we obtain
dt
dva
2
2attvxx iif
dtadv dt
dxv vdtdx
Graphical representation
Graphical representation
Graphical representation
Graphical representation
Graphical representation
Graphical representation
Graphical representation
Graphical representation
Graphical representation
Graphical integration
f
i
t
t
if dttvxx )(
dt
dxv
0lim ( )
f
in
t
xn n xttn
v t v t dt
Graphical integration
f
i
t
t
if dttvxx )(
dt
dxv
f
i
t
t
if dttavv )(
dt
dva
Answers to the even-numbered problems
Chapter 2
Problem 4:
(a) 50.0 m/s (b) 41.0 m/s
Answers to the even-numbered problems
Chapter 2
Problem 6:
(a) 27.0 m(b) 27.0 m + (18.0 m/s)∆t + (3.00 m/s2)(∆t)2
(c) 18.0 m/s
Answers to the even-numbered problems
Chapter 2
Problem 12:
(b) 1.60 m/s2; 0.800 m/s2
Answers to the even-numbered problems
Chapter 2
Problem 20:
(a) 6.61 m/s(b) −0.448 m/s2
Answers to the even-numbered problems
Chapter 2
Problem 38:
1.79 s
Answers to the even-numbered problems
Chapter 2
Problem 48:
(b) 3.00 × 10−3 s (c) 450 m/s (d) 0.900 m