02 Impulse & Momentum

8/9/2019 02 Impulse & Momentum

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MOMENTUM(MOMENTUM(

p

p))

%f a particle is mo#ing oliquel! and has #elocit! components v&,

v!, v', then its momentum components would e:

x x mv p = y y mv p = z z mv p =


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dt pd F

=

IMPULSE(IMPULSE(

J J) - MOMENTUM() - MOMENTUM(

p

p) THEOREM) THEOREM

v1 = 0v2 > 0

∫

∑

=

2

(

2

(

p

p

t

t pd dt F

⇒

∫

∑

dt F

t 1 t 2

∑

F F ∑

F

p p pdt F

t

t

∫ ∑

(2

2

(

The change in momentum is

affected ! the applied net forceand how long it is applied

⇒

%mpulse ( J ), whose

direction is the same

as the net force.

p J ⇒

Impulse – Momentum

Theorem

The change in momentum of a particle during a time inter#al

equals the impulse of the net

force that acts on the particle

during that inter#al

⇒


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IMPULSE(IMPULSE(


p

p) THEOREM) THEOREM

%n specific prolems, it is often easiest to use %mpulse*Momentum

Theorem in its component form:

x x p J

x x

t

t x p pdt F

(2

2

(

∫

∑

x x

t

t x mvmvdt F (2

2

(

∫ ∑

y y p J

y y

t

t y p pdt F

(2

2

(

∫

∑

y y

t

t ymvmvdt F (2

2

(

∫ ∑

z z p J

z z

t

t z p pdt F (22

(

∫ ∑

z z

t

t z mvmvdt F

(2

2

(

∫

∑


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IMPULSE(IMPULSE(


p

p) THEOREM) THEOREM

+uppose !ou ha#e a choice etween catching a .-*g all

mo#ing at /. m0s or a .*g all mo#ing at 2 m0s. 1hich will

e easier to catch

Example 1.

$ .3*g hoce! puc is mo#ing on an ic!, frictionless,

hori'ontal surface. $t t 4 the puc is mo#ing to the right at 5.

m0s. a) 6alculate the #elocit! of the puc(magnitude and

direction) after a force of 2-. N directed to the right has een

applied for .- s. ) %f instead, a force of 2 N directed to theleft is applied from t 4 s to t 2 4 .- s, what is the final

#elocit! of the puc

Example 2.


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CONSERVATION OF MOMENTUMCONSERVATION OF MOMENTUM

Before Collision

Durin Collision

m! mB

! B

!fter Collision

v! vB

m!

u!

mB

uB

F !B F B!

BA AB F F " # t

t F t F BA AB

BA AB J J

= B p A p

=

(2 B B p p (2 A A p p

22(( B A B A p p p p

The total momentum of a s$stem is

%onser&e'(

B B A A B B A A umumvmvm


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CONSERVATION OF MOMENTUMCONSERVATION OF MOMENTUM

Before Collision Durin Collision !fter Collision

Bx B Ax A Bx B Ax A vmvmumum

By B Ay A By B Ay A vmvmumum

Bz B Az A Bz B Az A vmvmumum

v!$

v!x

v!$

v!x

u!x

u!$

uB$

uBx


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COLLISIONCOLLISION

Elasti% Collision ) the total inetic energ! of the s!stem efore

and after the collision is constant.

2222

2

(

2

(

2

(

2

(


22(( B A B A K K K K

2222

2

(

2

(

2

(

2

( B B B B A A A A vmumumvm

2222

B B B A A A vumuvm

( .eqnvuvumuvuvm B B B B B A A A A A

⇒


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COLLISIONCOLLISION

6onser#ation of Momentum

B B B B A A A A vmumumvm

2 .eqnvumuvm B B B A A A ⇒


B B B

B B B B B

A A A

A A A A A

vum

vuvum

uvm

uvuvm

=

B B A A vuuv

A B B A uuvv

eqn. 7 eqn. 2

B A B A uuvv

⇒

B A vv

⇒ B A uu

Relati#e #elocit! of approach

Relati#e #elocit! of separation


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COLLISIONCOLLISION

+o, 6oefficient of Restitution, e for 8lastic 6ollision:

(

=

B A

B Avv

uue

*ummar$+

Elasti% Collision

9 K 4 K 29 e 4

Inelasti% Collision

9 K ; K 29 e <

Completel$

Inelasti% Collision

9 K ; K 29 e 4

9 u$ 4 u=


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COLLISIONCOLLISION

$ -*gm ullet is fired from a /*g gun. The mu''le speed is 3m0s. >ind the speed of recoil of the gun.

Example ,.

$ *gm loc mo#ing with a speed of cm0s to the right

collides with a 2*g loc mo#ing with a speed of - cm0s at thesame direction. a) >ind the speed of the two locs right after the

collision if the collision is inelastic with coefficient of restitution,

e 4 .?-. ) >ind the change in inetic energ! of the s!stem.

Example -.

$ hoce! puc B rests on a smooth ice surface and is struc ! a

second puc A, which was originall! tra#elling at / m0s and

which is deflected 5o from its original direction. @uc B

acquires a #elocit! at a /-o angle to the original direction of A.

The pucs ha#e the same mass. 6ompute the speed of each puc

after the collision.

Example .


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02 Impulse & Momentum

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