A ball of mass 0.5kg moving with a velocity of 10ms-1 makes a head on collision with a ball B of mass 2kg moving with a velocity of 9ms-1 in the opposite direction . If A and B stick together after collision , calculate the common velocity in the direction of A .
By conservation of momentum:
m1v1 + m2v2 = (m1 + m2)vf
where m1 and v1 are the mass and velocity of ball A before the collision, m2 and v2 are the mass and velocity of ball B before the collision, and vf is the common velocity of the two balls after the collision.
Plugging in the given values:
(0.5 kg)(10 m/s) + (2 kg)(-9 m/s) = (0.5 kg + 2 kg)vf
5 kg m/s - 18 kg m/s = 2.5 kg vf
-13 kg m/s = 2.5 kg vf
vf = -5.2 m/s
Since the question asks for the common velocity in the direction of A, we take the magnitude of vf (5.2 m/s) but add a positive sign:
Common velocity in direction of A = 5.2 m/s
To solve this problem, we can apply the principle of conservation of linear momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
1. Determine the initial momentum of Ball A (mA) and Ball B (mB):
Momentum of Ball A (mA) = mass of A (mA) * velocity of A (vA)
= 0.5 kg * 10 m/s
= 5 kg·m/s
Momentum of Ball B (mB) = mass of B (mB) * velocity of B (vB)
= 2 kg * (-9 m/s) (opposite direction)
= -18 kg·m/s (negative due to opposite direction)
2. Determine the total initial momentum before the collision:
Initial momentum before the collision = mA + mB
= 5 kg·m/s - 18 kg·m/s (subtracting due to opposite directions)
= -13 kg·m/s
3. Determine the total mass of both balls after collision:
Total mass after collision = mass of A + mass of B
= 0.5 kg + 2 kg
= 2.5 kg
4. Determine the final velocity after the collision (vfinal):
Final momentum after the collision = total mass after collision * vfinal
Because the balls stick together after the collision, we can substitute the combined mass of the balls (2.5 kg) for the total mass after collision:
Final momentum after the collision = 2.5 kg * vfinal
Since the final momentum after the collision should be equal to the initial momentum before the collision (conservation of linear momentum), we can set up an equation:
2.5 kg * vfinal = -13 kg·m/s
5. Solve for the final velocity (vfinal):
vfinal = -13 kg·m/s / 2.5 kg
= -5.2 m/s (taking negative value due to the opposite direction)
Therefore, the common velocity in the direction of A after the collision is 5.2 m/s.