Two objects of mass m and 2m are travelling in opposite directions with the same speed v. The objects collide and stick together. What is the kinetic energy lost in the collision?
conserving momentum,
mv - 2mv = 3mv'
v' = -v/3
KE lost is thus
(1/2 mv^2 + 1/2 * 2m (-v)^2) - 1/2 (3m)v'^2
...
Well, that collision sounds like a real "smashing" time! To determine the kinetic energy lost, let's break it down:
Before the collision, object 1 has a kinetic energy of (1/2)mv^2, and object 2 has a kinetic energy of (1/2)(2m)v^2.
After the collision, the two objects stick together (like Velcro buddies!) and form a single object with a mass of 3m.
So, the total kinetic energy after the collision is (1/2)(3m)v^2, as the final object is moving with the combined velocity of the two initial objects.
The kinetic energy lost in the collision would be the difference between the initial and final kinetic energies:
Kinetic energy lost = (1/2)mv^2 + (1/2)(2m)v^2 - (1/2)(3m)v^2
Simplifying that, we get:
Kinetic energy lost = (1/2)mv^2 - (1/2)mv^2
And what does that equal? Zero! That's right, in this collision, no kinetic energy is lost! It's as if the universe is a huge fan of conservation of energy and thinks everything should stay balanced.
So, grab your popcorn and enjoy the show, because the kinetic energy is staying put in this collision.
To find the kinetic energy lost during the collision, we need to compare the kinetic energy before and after the collision.
Before the collision, the two objects are moving in opposite directions but with the same speed v. The kinetic energy of an object is given by the equation KE = (1/2)mv^2, where m is the mass and v is the velocity.
So, the total kinetic energy before the collision is given by:
KE_before = (1/2)m(v^2) + (1/2)(2m)(v^2) = (1/2)mv^2 + mv^2 = (1 + 2)m(v^2) = 3mv^2
After the collision, the two objects stick together. Since they stick together, their final velocity will be the same, which we can call V.
The total mass after the collision is the sum of the masses of the two objects: m + 2m = 3m.
Using the equation for kinetic energy, the total kinetic energy after the collision is given by:
KE_after = (1/2)(3m)(V^2) = (3/2)m(V^2)
The kinetic energy lost during the collision is the difference between the kinetic energy before and after the collision:
KE_lost = KE_before - KE_after
Substituting the values we found:
KE_lost = 3mv^2 - (3/2)m(V^2)
Dividing both sides of the equation by m, we get:
KE_lost = 3v^2 - (3/2)V^2
So, the kinetic energy lost in the collision is 3v^2 - (3/2)V^2.
To find the kinetic energy lost in the collision, we need to calculate the initial kinetic energy of the system before the collision and compare it to the final kinetic energy after the collision.
Let's begin by calculating the initial kinetic energy of the system before the collision. The kinetic energy of an object is given by the equation:
KE = (1/2) * m * v²
For the first object with mass m, its initial kinetic energy is KE1 = (1/2) * m * v².
For the second object with mass 2m, its initial kinetic energy is KE2 = (1/2) * (2m) * v² = 2 * (1/2) * m * v² = m * v².
Therefore, the total initial kinetic energy of the system is the sum of the individual initial kinetic energies:
KE_initial = KE1 + KE2 = (1/2) * m * v² + m * v² = (1/2 + 1) * m * v² = (3/2) * m * v².
Now, let's consider the final kinetic energy of the system after the collision. The two objects collide and stick together, which means they combine to form a single object with mass 3m (m + 2m). Since they stick together, they move with the same velocity after the collision.
The final kinetic energy of the combined object is therefore given by:
KE_final = (1/2) * (3m) * V^2
To find the velocity V after the collision, we can use the principle of conservation of momentum. The total momentum before the collision is zero since the objects are moving in opposite directions with equal speeds. Therefore, the total momentum after the collision must also be zero.
The initial momentum before the collision is given by:
momentum_initial = m * (-v) + 2m * v = -m * v + 2m * v = m * v.
The final momentum after the collision is given by:
momentum_final = (3m) * V.
Using the conservation of momentum:
momentum_initial = momentum_final
m * v = (3m) * V
V = v/3
Now we can calculate the final kinetic energy:
KE_final = (1/2) * (3m) * (v/3)^2 = (1/2) * (3m) * (v^2/9) = (1/6) * m * v^2.
The kinetic energy lost in the collision can be calculated by subtracting the final kinetic energy from the initial kinetic energy:
KE_lost = KE_initial - KE_final = (3/2) * m * v² - (1/6) * m * v² = (9/6 - 1/6) * m * v² = (8/6) * m * v² = (4/3) * m * v².
Therefore, the kinetic energy lost in the collision is given by (4/3) * m * v².