a 2.0 kg block moving at 8 m/s on a frictionless surface collides elastically with a block at rest. The first block moves in the same direction at 2 m/s. What is the second block's mass?
To solve this problem, we can use the principles of conservation of momentum and kinetic energy in an elastic collision.
First, let's define the given information:
Mass of the first block (block in motion) = 2.0 kg
Velocity of the first block before the collision = 8 m/s
Final velocity of the first block after the collision = 2 m/s
Mass of the second block (block at rest) = m (unknown)
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:
(mass of first block × velocity of first block before collision) = (mass of first block × velocity of first block after collision) + (mass of second block × velocity of second block after collision)
Applying these values,
(2.0 kg × 8 m/s) = (2.0 kg × 2 m/s) + (m kg × vf)
16 kg·m/s = 4 kg·m/s + m kg·vf
Now, let's use the principle of conservation of kinetic energy for an elastic collision. In an elastic collision, both momentum and kinetic energy are conserved.
Kinetic energy before collision = Kinetic energy after collision
(1/2 × mass of first block × velocity of first block before collision^2) = (1/2 × mass of first block × velocity of first block after collision^2) + (1/2 × mass of second block × velocity of second block after collision^2)
Applying these values,
(1/2 × 2.0 kg × (8 m/s)^2) = (1/2 × 2.0 kg × (2 m/s)^2) + (1/2 × m kg × (vf)^2)
(1/2 × 2.0 kg × 64 m^2/s^2) = (1/2 × 2.0 kg × 4 m^2/s^2) + (1/2 × m kg × (vf)^2)
64 kg·m^2/s^2 = 8 kg·m^2/s^2 + m kg·(vf)^2
Now we have two equations:
16 kg·m/s = 4 kg·m/s + m kg·vf
64 kg·m^2/s^2 = 8 kg·m^2/s^2 + m kg·(vf)^2
To solve for the mass of the second block (m), we need to solve these two equations simultaneously. Substituting the value of vf from the first equation into the second equation, we get:
64 kg·m^2/s^2 = 8 kg·m^2/s^2 + m kg·((16 kg·m/s - 4 kg·m/s)/m)^2
Simplifying this equation, we get:
64 kg·m^2/s^2 = 8 kg·m^2/s^2 + m kg·(12 kg·m/s)^2
64 kg·m^2/s^2 = 8 kg·m^2/s^2 + m kg·(144 kg^2·m^2/s^2)
64 kg·m^2/s^2 = 8 kg·m^2/s^2 + 144 m kg·(kg·m^2/s^2)
Simplifying further:
64 kg·m^2/s^2 = 8 kg·m^2/s^2 + 144 m kg·m^2/s^2
64 kg·m^2/s^2 = 8 kg·m^2/s^2 + 144 m kg·m^2/s^2
Subtracting 8 kg·m^2/s^2 from both sides:
56 kg·m^2/s^2 = 144 m kg·m^2/s^2
Dividing both sides by 144 kg·m^2/s^2:
56/144 = m
Simplifying:
m ≈ 0.38 kg
Therefore, the mass of the second block is approximately 0.38 kg.
To solve this problem, we can use the principle of conservation of momentum, which states that the total momentum before and after a collision remains constant, provided there is no external force acting on the system.
Let's consider the first block as Block 1 with a mass of 2.0 kg and an initial velocity of 8 m/s, and the second block as Block 2 with an unknown mass (m2) and an initial velocity of 0 m/s.
The momentum before the collision can be calculated by multiplying the mass of each block by their respective velocities:
Initial momentum of Block 1 (P1) = mass of Block 1 (m1) * velocity of Block 1 (v1)
= 2.0 kg * 8 m/s
= 16 kg·m/s
Initial momentum of Block 2 (P2) = mass of Block 2 (m2) * velocity of Block 2 (v2)
= m2 * 0 m/s
= 0 kg·m/s
Since there is no external force, the total momentum before the collision is equal to the total momentum after the collision. Since the collision is elastic, the total momentum is conserved:
Total initial momentum = Total final momentum
P1 + P2 = P1' + P2'
Substituting the known values into the equation:
16 kg·m/s + 0 kg·m/s = (2.0 kg * 2 m/s) + (m2 * v2)
Since Block 1 moves in the same direction at 2 m/s, the final velocity of Block 1 will also be 2 m/s. Therefore, we can substitute that value into the equation and solve for the mass of Block 2 (m2).
16 kg·m/s = (2.0 kg * 2 m/s) + (m2 * 2 m/s)
16 kg·m/s = 4 kg·m/s + 2 m/s * m2
Subtracting 4 kg·m/s from both sides of the equation:
12 kg·m/s = 2 m/s * m2
Dividing both sides of the equation by 2 m/s:
6 kg = m2
Therefore, the mass of the second block is 6 kg.