Question

Calculate the magnitude of the velocity and the θ angular direction of the block and the bullet together when the 50 g bullet moves to sky at a speed of 600 m/s and hits a sliding 4 kg block with a speed of 12 m/s in a direction with 30o with respect to ground.

Answer 1

Answer:

Magnitude of the velocity = 16.82 m/s

Angular direction, θ = 52.41°

Explanation:

As given ,

mass of bullet, m₁= 50g = 0.05 kg

speed of bullet , u₁ = 600 m/s

mass of the block , m₂ = 4 kg

speed of the block before collision , u₂ = 12 m/s

direction , θ = 30°

Now,

Assume that the combined velocity of bullet and block after collision = v

and the direction = θ

Now, from the conservation of momentum in x - direction :

m₁ u₁ + m₂ u₂ = ( m₁ + m₂ ) vₓ

where v = final velocity after collision

u₁ = initial velocity of bullet before collision = 0

m₁ = mass of the bullet before collision = 0.05 kg

u₂  = velocity of block before collision = 12 cos(30° )

m₂ = mass of block before collision

m₁ + m₂ = combined mass of bullet and block after collision = 0.05 + 4

∴ we get

0.05 (0) + 4(12 cos(30° ) ) = ( 0.05 + 4 ) vₓ

⇒ 0 + 4(6√3) = 4.05 vₓ

⇒24√3 = 4.05 vₓ

⇒vₓ = 10.26 m/s

Now, from the conservation of momentum in y - direction :

m₁ u₁ + m₂ u₂ = ( m₁ + m₂ ) $v_{y}$

where v = final velocity after collision

u₁= initial velocity of bullet before collision = 600

m₁ = mass of the bullet before collision = 0.05 kg

u₂  = velocity of block before collision = 12 sin(30° )

m₂= mass of block before collision

m₁+ m₂= combined mass of bullet and block after collision = 0.05 + 4

∴ we get

0.05 (600) + 4(12 sin(30° ) ) = ( 0.05 + 4 ) $v_{y}$

⇒ 30 + 4(6) = 4.05 $v_{y}$

⇒30 +24 = 4.05 $v_{y}$

⇒54 = 4.05 $v_{y}$

⇒$v_{y}$ = 13.33 m/s

Now, the magnitude of the velocity = √vₓ² + $v_{y}$² = √(10.26)² + (13.33)²

                                                           = √105.26 + 177.68

                                                           = √282.95 = 16.82

The angular direction, θ =  $tan^{-1}$($\frac{v_{y} }{v_{x} }$) =  $tan^{-1}$($\frac{13.33}{10.26}$) = $tan^{-1}$(1.299) = 52.41°