The two blocks of masses M and 2M shown above initially travel at the same speed v but in opposite directions. They collide and stick together. What is the final velocity of the mass M after the collision?
negative one sixth v.
negative one third v.
Zero
one sixth v.
one third v.
Initial momentum must equal final momentum
V = initial velocity of the blocks, v = final velocity of the two
2VM – VM = 3Mv
MV = 3Mv
v = ⅓V
initial KE = ½MV² + ½2MV² = 1.5MV²
final KE = ½3Mv²
Lost KE = 1.5MV² – ½3Mv²
substituting v = ⅓V
Lost KE = 1.5MV² – ½3M(⅓V)²
Lost KE = 1.5MV² – ½3(1/9)MV²
Lost KE = 1.5MV² – (1/6)MV²
Lost KE = 1.333MV²
and that is (d)
(D) 4/3 Mv2
Well, isn't that a collision with some "mass appeal"! When these blocks collide and stick together, we need to take a closer look at the "M and 2M" duo. Since they're traveling in opposite directions, their total momentum is initially zero, which means the final momentum must also be zero. Since the final velocity is related to the total momentum, we can conclude that the final velocity of the mass M will be zero. So, the correct answer here is Z-E-R-O! Keep on rolling with the physics, my friend!
To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is defined as the product of its mass and velocity. In this case, the total momentum before the collision is given by:
Initial momentum = (mass M) * velocity v - (mass 2M) * velocity v
= Mv - 2Mv
= -Mv
After the collision, the blocks stick together and move as one object. Let's call this final velocity V. The total momentum after the collision is given by:
Final momentum = (mass M + mass 2M) * final velocity V
= 3M * V
Since the total momentum before the collision is equal to the total momentum after the collision, we can set these two values equal to each other:
-Mv = 3M * V
Simplifying this equation, we find:
Mv = -3M * V
Now, we can solve for V by dividing both sides of the equation by -3M:
V = -Mv / (3M)
Simplifying further, we find:
V = -v / 3
Therefore, the final velocity of the mass M after the collision is negative one third (1/3) of the initial velocity v.
So, the correct answer is: negative one third v.
To find the final velocity of the mass M after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
In this case, let's assume the initial velocity of the mass M is v1 and the initial velocity of the mass 2M is v2. Since they have the same speed v but in opposite directions, we can write v1 = -v and v2 = v.
The momentum before the collision is given by P1 = M * v1 + 2M * v2. Substituting the values, we have P1 = M * (-v) + 2M * v = -M * v + 2M * v = M * v.
After the collision, the masses stick together, so we can consider them as a single combined mass. Let's assume the final velocity of this combined mass is vf. The momentum after the collision is given by P2 = (M + 2M) * vf = 3M * vf.
Since the total momentum before the collision (P1) should be equal to the total momentum after the collision (P2), we have M * v = 3M * vf. Simplifying, we get vf = (M * v) / (3M) = v / 3.
Therefore, the final velocity of the mass M after the collision is one third (1/3) of the initial velocity v. So the correct answer is "one third v".