Two blocks can collide in a one-dimensional collision. The block on the left has

Question

Two blocks can collide in a one-dimensional collision. The block on the left hass a mass of 0.50 kg and is initially moving to the right at 2.4 m/s toward a second block of mass 0.50 kg that is initially at rest. When the blocks collide, a cocked spring releases 1.2 J of energy into the system. (For velocities, use + to mean to the right, - to mean to the left.) What is the velocity of the first block after the collision?___m/s What is the velocity of the second block after the collision?___m/s Remember that the blocks cannot pass through each other!

Explanation / Answer

we are talking non-elastic collision, WE need to compute using the moments. use the spring's energy to accelerate both blocks

energy before spring's action = (m1+m2)v²/2
energy after spring's action = (m1+m2)v²/2 + 1.2J

So speed of the two blocks is then the solution of

(m1+m2)v²/2 + 1.2J = (m1+m2)V²/2



u1 ... speed of block 1 before collision
v1 ... speed of block 1 after collision
v2 ... speed of block 2 after collision

From the conservation of momentums, we get

m1u1 = m1v1 + m2v2

From the conservation of energy + the added 1.2J (which we use only for moving things around - not for heating or scratching the blocks), we get

m1u1²/2 + 1.2 = m1v1²/2 + m2v2²/2

So we have two equations for the unknowns v1 and v2, which we can (try to) solve: From the first equation, we get

v2 = m1/m2*(u1-v1)

Using this in the second equation, we get

m1u1² + 2.4 = m1v1² + (m1*(u1-v1))²/m2

which is a quadratic equation for v1. Its solution gives the speed of v1. Putting this into the previous equation, we get v2.

Using the numbers, the equation to be solved is

0.5*2.4² + 2.4 = 0.5*v1² + 0.25*(2.4-v1)²/0.5
5.856 = 0.6*v1² + 0.72*(2.4-v1)²
5.856 = 0.6*v1² + 0.72*(2.4²-4.8*v1+v1²)
5.856 = 0.6*v1² + 4.1472 - 3.456*v1 + 0.72*v1²
1.7088 = 1.32*v1² - 3.456*v1
1.2945 = v1² - 2.6182*v1
3.0083 = v1² - 2.6182*v1 + 1.7137
3.0083 = (v1 - 1.3091)²
1.7344 = v1 - 1.3091
3.043 = v1
or
-1.7344 = v1 - 1.3091
-0.4253 = v1

Only the latter solution (where block1 now moves to the left) makes sense [but what is that other, positive solution??]

For v2, we'd get from

v2 = m1/m2*(u1-v1)

v2 = 0.5/0.5*(2.4-(-0.4253)) m/s = 1.2*(2.4+0.4253) m/s = 1.2*2.8253 m/s = 3.39 m/s.

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