Momentum question x2

(1) let m2 = mass of jet fighter = 8000kg and m1 = mass of missile = 20kg, v= velocity of jet fighter = 100m/s, v1 = velocity of missile = 500m/s, v2 = velocity of jet fighter shoot the first 5 missiles, v3 = velocity of jet figher shoot the second 5 missiles, m3 = mass of jet fighter shoot first 5 missiles = 8000-5 x 20 = 7900kg, m4 = mass of jet fighter shoot second 5 missiles = 7900 - 5 x20 = 7800kg, v4 = velocity of missile shoot missiles at the second time = -500m/s

(a) m2 v = 5m1v1 + m3 v2
8000 x 100 = 5 x 20 x500 + 7900 x v2
v2 = 94.937m/s (to the right)
(b) m3v2 = 5m1v4 + m4 v3
7900 x 94.937 = 5 x 20 x (-500) + 7800 x v4
v4 = 102.56m/s (to the right)

(2) let a = angle of the cart Y move at an angle to the original direction
let mx = mass of cart X and my = mass of cart Y
let v1 = initial velocity of cart X
let v2 = final velocity of cart X
let v3 = final velocity of cart Y
(a) consider the direction along original moving direction after collision
mx v1 = mx v2cos 45 + my v3 cos a
75 (2) = (75)(1.2) cos 45 + 150 v3 cos a
v3 cos a = 0.576 -----------(1)

consider the direction perpendicular to the moving direction afetr collision
mx v2 sin 45 + 150 v3 sin a = 0
75(1.2) sin 45 + 150 v3 sin a = 0
v3 sin a = -0.424 -------------(2)
Eq(2)/eq(1) : tan a = -0.736
angle a = -36.387
sub into (2): final velocity of cart Y, v3 = 0.715m/s at angle -36.387 to the original direction

2010-03-02 10:10:38 補充：
sin a is -ve value.
cos a is positive value.
The angle a is in the fourth quadrant.
If the x-direction towards right is +ve and y-direction towards up is +ve.

The cart Y will moves towards right with 36.387 degree to the original direction.

1. (a) Using conservation of momentum,
Momentum before firing of missiles = 8000 x 100 Ns
Momentum after firing of missiles = [(8000-5x20)v + (5x200) x 500] Ns
where v is the speed of the fighter after missile firing
hence, 800 000 = (8000-5x20)v + (5x200) x 500
solve for v gives v = 94.9 m/s

(b) Similarly, using comservation of momentum
(8000 - 5 x 20) x 94.9 = [(8000 - 10x20)v' + (5x20)(-500)]
where v' is the speed of the fighter after firing further 5 missiles to the left
solve for v' gives v' = 103 m/s

2. Let v be the final velocity of cart Y after collision, making an angle a with the initial moving direction of cart X
Resolve the final momentum of cart X and Y along the original direction of cart X, we have, using conservation of momentum,

75 x 2 = 75 x 1.2cos(45) + 150v.cos(a) ------------------- (1)

Resolve momentum in the direction perpendicular to the original direction of cart X, using conservation of momentum,

75 x 1.2cos(45) = 150v.sin(a) ----------------- (2)

from (1), 150v.cos(a) = 70.45 ---------------------- (3)
(2)/(3): tan(a) = 75x1.2cos(45)/70.45
a = arc-tan[75x1.2cos(45)/70.45]

Then use the value of angle a to calculate v from equation (3)