Glancing Collisions
Glancing Collisions
„
the total momentum of the system in each
direction is conserved
m1v 1ix + m2 v 2ix = m1v 1fx + m2 v 2 fx and
ƒ
„
„
„
pi = pf => pix = pfx and piy = pfy
Momentum is conserved in the x direction and
in the y direction
Apply conservation of momentum separately
to each direction
Kinetic energy is conserved for glancing
elastic collisions too. Remember kinetic
energy is a scalar so the KE conservation
equation is the same for glancing and
head-on collisions.
m1v 1iy + m2 v 2iy = m1v 1fy + m2 v 2 fy
For the figure above:
m1v1i = m1v1f cos θ + m2v2f cos φ (x-component)
0 = m1v1f sin θ - m2v2f sin φ (y-component)
The Radian
„
„
1
1
1
1
m1v12i + m2 v22i = m1v12f + m2 v22 f
2
2
2
2
v1i + v1f = v2i + v2f
More About Radians
„
„
Comparing degrees and radians
„
360°
= 57.3°
2π
Converting from degrees to
radians
„
θ [rad ] =
π
180°
θ [deg rees] =
1
θ [deg rees]
57.3°
θ =
s
r
o
57.3
Angular Displacement
1 rad =
„
The radian is a
unit of angular
measure
The radian can be
defined as the arc
length s along a
circle divided by
the radius r
„
Axis of rotation is
the center of the
disk
Need a fixed
reference line
During time t, the
reference line
moves through
angle θ
1
Average Angular Speed
„
The average
angular speed, ω,
of a rotating rigid
object is the ratio
of the angular
displacement to
the time interval
ω av =
θf − θi
tf − ti
=
„
Relationship Between Angular
and Linear Quantities
Displacements
„
Speeds
„
Accelerations
„
∆s = ∆θr
vt = ω r
at = α r
The average angular acceleration α
of an object is defined as the ratio
of the change in the angular speed
to the time it takes for the object
to undergo the change:
α av =
∆θ
∆t
„
Average Angular
Acceleration
„
Every point on
the rotating
object has the
same angular
motion
Every point on
the rotating
object does not
have the same
linear motion
ωf − ωi
tf − ti
=
∆ω
∆t
Rotational kinematic equations
vt = vti + att
r
r
r
=> ω = ωi + αt
Similarly:
ω2 = ωi2 + 2α ∆θ
∆θ = ωit + (1/2) αt2
Analogies Between Linear
and Rotational Motion
2