The mass of the blue puck shown below is 10.0% greater than the mass of the green one. Before colliding, the pucks approach each other with momenta of equal magnitudes and opposite directions, and the green puck has an initial speed of 8.0 m/s. Find the speeds of the pucks after the collision if half the kinetic energy of the system becomes internal energy during the collision.
The picture shows the two pucks going towards each other in a straight line (blue goes left and green goes right)and then after they hit the travel in opposite directions at 30 degree angles (blue goes down green goes up).
Why do they give an initial velocity? Shouldn't the initial momentum of the system be zero because they're going in opposite directions and equal magnitude? If this is true, then how can you solve for final velocity?
The initial momentum is zero, so the final momentum has to be zero. Now on final momentum, yes it has to equal zero, and frankly, I am not certain what you mean by "after they hit they travel in opposite directions at 30 degree angles" UNLESS you mean one is going to the right upwards at 30 deg, and one is going LEFT down 30 deg.
IF this is so, then THe upwards momentum of one (Massblue*vblue*sin30+ plus the upward momentum of the other (massgreen*vgreen*sin-30) is zero. Now you have one other equation:
horizontal momenum:
the sum of those is zero.
You have two equations, two unknowns.
Something actually seems wrong to me with the problem, as you pointed out.
The initial momentum of the system is indeed zero because the pucks have equal magnitudes of momentum but in opposite directions. However, the initial velocity of the green puck is provided in order to find the final velocities of both pucks after the collision.
To solve for the final velocities, we need to consider the conservation of momentum and the conservation of kinetic energy. The conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. In this case, since the initial momentum is zero, the total momentum after the collision is also zero.
Let's denote the mass of the green puck as m, and the mass of the blue puck as 1.10m (since it is stated that the blue puck's mass is 10% greater than the green puck's mass).
Using these notations, the initial momentum of the green puck is m * 8.0 m/s = 8m, and the initial momentum of the blue puck is -1.10m * v (where v is the magnitude of its velocity since it goes to the left).
Since the total momentum after the collision is zero, we can write the equation:
8m - 1.10m * v - 1.10m' * v' = 0,
where m' and v' are the mass and magnitude of the velocity of the blue puck after the collision.
Now, to consider the conservation of kinetic energy, it is given that half of the kinetic energy of the system becomes internal energy during the collision. The formula for kinetic energy is 1/2 * mass * velocity^2. Using this, we can write the following equation:
(1/2 * m * (8.0)^2) / 2 = (1/2 * 1.10m * v^2) / 2 + (1/2 * m' * v'^2) / 2,
which simplifies to:
16m = 0.55m * v^2 + 0.55m' * v'^2.
We have two equations (momentum and kinetic energy) with two unknowns (v and v'). Solving these equations simultaneously will give us the values of v and v', which represent the speeds of the green and blue pucks after the collision.
Please note that this is a simplified explanation. In reality, collisions can be more complex and involve factors such as rotation. However, for the purpose of this question, we are assuming a simplified scenario.
The initial velocity of the green puck is given to determine the initial momentum of the system. Although the pucks have equal magnitudes but opposite directions of momentum, the initial momentum of the system is not necessarily zero because the masses of the pucks are different.
To solve for the final velocities of the pucks, we can use the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant before and after a collision.
Before the collision, the initial momentum of the system can be calculated as the product of the mass and velocity of the green puck:
Initial momentum of the system = mass of the green puck * velocity of the green puck
Since the pucks have opposite directions of momentum, the initial momentum of the system will be negative.
After the collision, the final momentum of the system is also the sum of the individual momenta of the pucks:
Final momentum of the system = momentum of the green puck + momentum of the blue puck
Since the pucks move in opposite directions after the collision, the directions of their momenta will also be opposite.
By equating the initial and final momenta of the system, we can solve for the final velocities of the pucks. However, since the collision is not elastic and some kinetic energy is lost, we also need to take into account the change in kinetic energy.
Let's denote the initial mass of the green puck as m, and the mass of the blue puck as 1.1m (10% greater than the green puck).
Let's also denote the initial velocity of the green puck as v.
Initial momentum of the system = mv
Final momentum of the system = -mv' + (1.1m)(-v')
Since half the kinetic energy of the system becomes internal energy, we can write:
Change in kinetic energy = (1/2)mv^2 - (1/2)mv'^2
By equating the initial and final momenta, and considering the change in kinetic energy, we can solve for the final velocities of the pucks.