Two blocks move along a linear path on a nearly frictionless air track. One block, of mass 0.107 kg, initially moves to the right at a speed of 4.70 m/s, while the second block, of mass 0.214 kg, is initially to the left of the first block and moving to the right at 6.80 m/s. Find the final velocities of the blocks, assuming the collision is elastic.
a.) Velocity of the 0.107 kg block= ?
b.) Velocity of the 0.214 kg block= ?
v₁₀=4.7 m/s, m₁=0.107 kg;
v₂₀=6.8 m/s, m₂=0.214 kg
v₁=? v₂=?
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v₁= {2m₂v₂₀ +(m₁-m₂)v₁₀}/(m₁+m₂)
v₂={ 2m₁v₁₀ + (m₂-m₁)v₂₀}/(m₁+m₂)
To find the final velocities of the blocks after an elastic collision, we can use the principle of conservation of momentum.
The momentum before the collision can be calculated as the product of mass and velocity for each block:
Initial momentum of the 0.107 kg block = (0.107 kg) * (4.70 m/s)
Initial momentum of the 0.214 kg block = (0.214 kg) * (-6.80 m/s) [negative because it is moving to the left]
Now, let's calculate the total initial momentum:
Total initial momentum = Initial momentum of the 0.107 kg block + Initial momentum of the 0.214 kg block
To find the final velocities, we need to use the formula for conservation of momentum:
Total initial momentum = Total final momentum
Since the collision is elastic, the total final momentum will be the negative of the total initial momentum.
Now, let's solve for the final velocities of the blocks:
Total initial momentum = Total final momentum
(0.107 kg) * (4.70 m/s) + (0.214 kg) * (-6.80 m/s) = -[(0.107 kg) * (v1) + (0.214 kg) * (v2)]
Simplifying the equation:
(0.107 kg) * (4.70 m/s) - (0.214 kg) * (6.80 m/s) = -[(0.107 kg) * (v1) + (0.214 kg) * (v2)]
Now, let's solve for v1 and v2, the final velocities of the blocks:
(0.107 kg) * (v1) + (0.214 kg) * (v2) = -[(0.107 kg) * (4.70 m/s) - (0.214 kg) * (6.80 m/s)]
Simplifying the equation:
0.107 kg * v1 + 0.214 kg * v2 = -0.107 kg * 4.70 m/s + 0.214 kg * 6.80 m/s
Now, let's calculate the final velocities of the blocks using the equation above.
To find the final velocities of the blocks after collision, we can use the law of conservation of momentum and the law of conservation of kinetic energy for an elastic collision.
The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be written as:
(initial momentum of block 1 + initial momentum of block 2) = (final momentum of block 1 + final momentum of block 2)
The momentum, p, of an object is defined as the product of its mass, m, and velocity, v. Therefore, we can rewrite the equation as:
(mass of block 1 * initial velocity of block 1 + mass of block 2 * initial velocity of block 2) = (mass of block 1 * final velocity of block 1 + mass of block 2 * final velocity of block 2)
For the given problem, let's denote:
m1 = mass of block 1 = 0.107 kg
v1i = initial velocity of block 1 = 4.70 m/s
m2 = mass of block 2 = 0.214 kg
v2i = initial velocity of block 2 = -6.80 m/s (negative because it is moving to the left)
Using the above equation, we can solve for the final velocities.
Step 1: Calculate the initial momenta of the blocks.
p1i = m1 * v1i
p2i = m2 * v2i
Step 2: Substitute the values into the equation.
(m1 * v1i + m2 * v2i) = (m1 * v1f + m2 * v2f)
Step 3: Rearrange the equation to solve for the final velocities.
(m1 * v1f + m2 * v2f) = (m1 * v1i + m2 * v2i)
m1 * v1f + m2 * v2f = m1 * v1i + m2 * v2i
v1f = (m1 * v1i + m2 * v2i - m2 * v2f) / m1
v2f = (m1 * v1i + m2 * v2i - m1 * v1f) / m2
Step 4: Substitute the known values and calculate the final velocities.
v1f = (m1 * v1i + m2 * v2i - m2 * v2f) / m1
v1f = (0.107 kg * 4.70 m/s + 0.214 kg * -6.80 m/s - 0.214 kg * v2f) / 0.107 kg
v2f = (m1 * v1i + m2 * v2i - m1 * v1f) / m2
v2f = (0.107 kg * 4.70 m/s + 0.214 kg * -6.80 m/s - 0.107 kg * v1f) / 0.214 kg
Now, you can plug in the values and calculate the final velocities of the blocks.