Notions of derivation and integration
The derivate of a function f (t) is defined as:
(t)
f � (t) = dfdt(t) = lim∆t→0 f (t+∆t)−f
.
∆t
Most important derivatives:
d log t
dt
1
n−1
=
nt
,
n
=
�
0;
=
dt
dt
t
df (t)
d sin t
d cos t
slope
tan
α(t)
=
= − sin t.
dt
dt = cos t,
dt
t
de
t
=
e
dt
�
�
=
f
(g(t))g
(t).
Derivate of a composite function: df (g(t))
dt
The integral
� � is the inverse function of the derivative
f (t) + C = f (t)dt.
n
Most important integrals:
� n
� 1
tn+1
t dt = n+1 + C, n �= −1; t dt = log t + C
�
�
cos tdt = sin t + C, sin tdt = −cost + C
� t
..
t
e dt = e + C .
Kinematics of the material point
�v
Ex: 2.1 pag 16
Mean speed (mean velocity) vm for the motion along
the x axis:
∆x
xf − xi
= [m
vm =
=
s ]
∆t
tf − ti
where: ∆x = space travelled by the point ∆t. xi , xf =
initial and final position, ti tf initial and final times.
Kinematics of the material point
Mean speed (mean velocity) vm for the motion along
the x axis:
∆x
xf − xi
m
vm =
=
=
[
s ]
∆t
tf − ti
where: ∆x = space travelled by the point ∆t. xi , xf =
initial and final position, ti tf initial and final times.
Instantaneous velocity
∆x
dx(t)
î =
î
�v = lim
∆t→0 ∆t
dt
Kinematics of the material point
Mean acceletion
am
∆v
vf − vi
=
=
∆t
t f − ti
=
m
[ s2 ]
where: ∆v = variation of velocity in the time interval ∆t.
vi , vf = velocity at the initial and final times, ti and tf .
Instantaneous acceleration
�
�
∆�v
d�v (t)
d d�x(t)
d2 �x(t)
�a = lim
=
=
=
∆t→0 ∆t
dt
dt
dt
d2 t
Uniform linear motion: �v = cost
x(t) = xi + v(t − ti )
x(t) and xi positions of the body (material point) at the
generic time t and initial time ti .
Constant acceleration: �a = cost
v(t) = vi + a(t − ti )
1
x(t) = xi + vi (t − ti ) + a(t − ti )2
2
v(t) e vi = velocity of the body at the times t and ti .
Free fall
The motion of a body (in the vacuum) subject to gravity
is uniformly accelerated downward, with acceleration
m
g ∼ 9.81 2
s
.
1 2
y(t) = h + vi t − gt
2
vy (t) = vi − gt
�
√
2h
fly time vi = 0: tf =
2gh.
g , final velocity: vf =
highest position: yM =
Ex: 2.6 pag 25
vi2
2g
at the time tM =
vi
g
Position vector and trajectory
The position vector �r(t) of an object w.r.t. a reference
frame is the vector that extends from the origin of the coordinates to the position of the object at time t. In 3D
�r(t) = x(t)î + y(t)ĵ + z(t)k̂
The displacement between position a time t1 , i.e. �r(t1 ) =
�r1 , and time t2 , i.e. �r(t2 ) = �r2 is
∆�r = �r2 − �r1
Motion in two (and three) dimensions
�v (t1 )
�a(t1 )�v (t2 )
Trajectory: The position of point in motion is de�a(t2 )
scribed at every time t by a position vector �r(t) = OP� (t),
�r(t2 )
�v (t3 )
with corresponding instantaneous velocity and accelerations,
�r(t1 )
�a(t3 )
d�
r (t)
d�
v (t)
�
r
(t
)
�v = dt e �a = dt . This allows for a decomposition of
3
the motion in independent components.
Motion in two (and three) dimensions
�v (t1 )
�a(t1 )�v (t2 )
Trajectory: The position of point in motion is de�a(t2 )
scribed at every time t by a position vector �r(t) = OP� (t),
�r(t2 )
�v (t3 )
with corresponding instantaneous velocity and accelerations,
�r(t1 )
�a(t3 )
d�
r (t)
d�
v (t)
�
r
(t
)
�v = dt e �a = dt . This allows for a decomposition of
3
the motion in independent components.
constant velocity: �v = cost, v =
x(t) = xi + vx (t − ti )
y(t) = yi + vy (t − ti )
{xi , yi } = initial position.
�
�v
vx2 + vy2
�v
�
Constant acceleration: �a = cost, a = a2x + a2y
vx (t) = vxi + ax (t − ti )
vy (t) = vyi + ay (t − ti )
x(t) = xi + vxi (t − ti ) + 12 ax (t − ti )2
y(t) = yi + vyi (t − ti ) + 12 ay (t − ti )2
{vi , vi } = initial velocity.
�a
�a
Projectile motion: with initial velocity �vi and angle
α w.r.t. x:
x(t) = (vi cos α)t
,
vx (t) = vi cos α
,
vi2 sin2 α
Height: yM =
,
2g
vi2 sin 2α
range xM =
.
g
1 2
y(t) = (vi sin α)t − gt
2
vy (t) = vi sin α − gt
α
fly time tf = 2 vi sin
, horizontal
g
It is the composition of a uniform linear motion in the orizzontal direction and of the uniformly
acceletared motion in the horizontal direction
Max horizontal range for 45
°
s(t) = Rθ(t)
Uniform circular motion
Angular velocity ω =
ω=
angle variation in ∆t
∆t
θ(t)
t
Period of the motion T
2π
ω=
T
Circular motion
angular velocity
ω=
dθ(t)
dt
speed (spatial speed)
v(t) =
ds(t)
dθ(t)
=R
= Rω
dt
dt
angular acceleration
α(t) =
dω(t)
dt
R
θ
angular velocity ω = [ rad
s ],
period T = 2π
ω = [s],
frequency f = T1 = [ rad
sec ] = [Hertz] = [Hz].
Uniform circular motion
s(t) = Rθ(t)
R
θ
Components:
x(t) = R cos ωt
dx(t)
vx (t) =
= −Rω sin ωt
dt
dvx (t)
ax (t) =
= −Rω 2 cos ωt
dt
,
,
,
y(t) = R sin ωt;
dy(t)
vy (t) =
= Rω cos ωt;
dt
dvy (t)
ay (t) =
= −Rω 2 sin ωt.
dt
Uniform circular motion
s(t) = Rθ(t)
R
Components:
θ
x(t) = R cos ωt
dx(t)
vx (t) =
= −Rω sin ωt
dt
dvx (t)
ax (t) =
= −Rω 2 cos ωt
dt
,
y(t) = R sin ωt;
dy(t)
vy (t) =
= Rω cos ωt;
dt
dvy (t)
ay (t) =
= −Rω 2 sin ωt.
dt
,
,
Ex: 4.9 pag.71
centripetal acceleration: the body moves on a circumference with constant velocity in mognitude. However
the velocity varies in direction from point to point, so that
the body is subject to an centripetal acceleration whose direction is toward the centre of the circumference:
v2
ac = ω R =
R
2
.
2
$x_{BA}$
Relative motion (1D)
The position xP A of P observed by A is the sum of the
position xP B of P observed by B, plus the position xBA of
B observed by A:
xP A = xP B + xBA
The velocity vP A of P w.r.t. A is the velocity vP B of P
w.r.t. B plus the relative velocity vBA of B w.r.t. A:
vP A = vP B + vBA
$x_
Relative motion (2D and 3D)
The position �rP A of P observed by A is the sum of the
position �rP B of P observed by B, plus the �rBA position
�rBA of B observed by A:
�rP A = �rP B + �rBA
.
The velocity �vP A of P w.r.t. A is the velocity �vP B of P
w.r.t. B plus the relative velocity �vBA of B w.r.t. A:
�vP A = �vP B + �vBA
If the velocity �vBA of B w.r.t. A is constant, the acceleration is the same for the two observers located in the
reference frames A and B.