Komponentform

[tosken]

Conservation of the x−component of total momentum gives

mAVA1x = mAVA2x + mBVB2x,
(0.500 kg)(4.00 m⁄s) = (0.500 kg)(2.00 m⁄s)(cos α) + (0.300 kg)(4.47 m⁄)(cos β),

and conservation of the y−component gives

0 = mAVA2y + mBVB2y,
0 = (0.500 kg)(2.00 m⁄s)(sin α) − (0.300 kg)(4.47 m⁄s)(sin β).

There are two simultaneous equations for α and β. The simplest solution is to eliminate β as follows: We solve the first equation for cos β and the second for sin β; we then square each equation and add. Since sin2 β + cos2 β = 1, this eliminates β and leaves an equation that can solve for cons α and hence for α. We can then substitute this value back into either of the two equations and solve the result for β. Try to figure out the details for yourself. I'll just give you the results here


α = 36.9°, β = 26.6°.

Hvordan kommer man frem til dette? Håper på svar.

Takk.

[Janhaa]

Conservation of the x−component of total momentum gives
mAVA1x = mAVA2x + mBVB2x,
(0.500 kg)(4.00 m⁄s) = (0.500 kg)(2.00 m⁄s)(cos α) + (0.300 kg)(4.47 m⁄)(cos β), (I)
and conservation of the y−component gives
0 = mAVA2y + mBVB2y,
0 = (0.500 kg)(2.00 m⁄s)(sin α) − (0.300 kg)(4.47 m⁄s)(sin β) (II).
There are two simultaneous equations for α and β. The simplest solution is to eliminate β as follows: We solve the first equation for cos β and the second for sin β; we then square each equation and add. Since sin2 β + cos2 β = 1, this eliminates β and leaves an equation that can solve for cons α and hence for α. We can then substitute this value back into either of the two equations and solve the result for β. Try to figure out the details for yourself. I'll just give you the results here
α = 36.9°, β = 26.6°.
Hvordan kommer man frem til dette? Håper på svar.
Takk.

[tex]\alpha\;:\;a[/tex]
[tex]\beta\;:\;b[/tex]

I: 2 = cos(a) + 1.34cos(b)

II: 1.34sin(b) = sin(a)

I gir: cos(b) = [2 - cos(a)] / 1.34
og
II gir: sin(b) = sin(a) / 1.34

Kvadrer begge sider:

I: [cos(b)][sup]2[/sup] = [2 - cos(a)][sup]2[/sup] / 1.34[sup]2[/sup]
og
II: [sin(b)][sup]2[/sup] = [sin(a)][sup]2[/sup] / 1.34[sup]2[/sup]

I + II gir III:

sin[sup]2[/sup]b + cos[sup]2[/sup]b = 1 = [4 - 4cos(a) + cos[sup]2[/sup]a + sin[sup]2[/sup]a] / 1.34[sup]2[/sup] (III)

husk at: cos[sup]2[/sup]a + sin[sup]2[/sup]a = 1

rydder opp etc i (III):

1.34[sup]2[/sup] = 5 - 4cos(a)

4cos(a) = 3.2

cos(a) = 0.8

[tex]\alpha\;=[/tex][tex]\;arc\;cos(0.8)\;=\;36.9^o[/tex]

II gir:

[tex]{sin(\beta )}[/tex][tex]\;=\;[/tex][tex]{sin(36.9^{o})}\over 1.34[/tex][tex]\;=0.448[/tex]

[tex]\beta\;=[/tex][tex]\;arc\;sin(0.448)\;=\;26.6^o[/tex]

[tosken]

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