Solving 2D Collision: Find v1 & v2 Magnitudes

[kbeyer]

A fireworks rocket is moving at a speed of 45.0 m/s. The rocket suddenly breaks into 2 pieces of equal mass, which fly off with velocities v1 and v2. v1 creates an angle of 30 degrees above the x-axis and v2 creates an angle of 60 degrees below the x-axis.

What is the magnitude of v1 and v2?

 

[St. Aegis]

Using the conservation of momentum, and this problem becomes equally easy because they give that the mass is the same;
Im not sure, are we supposed to solve these problems for other students? or help them learn

 

[baba89]

To solve for the magnitudes of v1 and v2, we can use the conservation of momentum and conservation of energy principles. The total momentum of the system before and after the collision must be equal, and the total kinetic energy must also be conserved.

First, we can calculate the initial momentum of the system by multiplying the mass of the rocket by its initial velocity of 45.0 m/s. This momentum must be divided equally between the two pieces after the collision, so each piece will have a momentum of half the initial momentum.

Next, we can use trigonometry to break down the velocities v1 and v2 into their x and y components. Since v1 creates an angle of 30 degrees above the x-axis, its x component will be v1x = v1*cos(30) and its y component will be v1y = v1*sin(30). Similarly, v2x = v2*cos(60) and v2y = v2*sin(60).

Using the conservation of momentum, we can set up the following equation:

m*v1 = m*v1x + m*v2x

We know that the mass of each piece is equal and that the initial momentum is divided equally between them, so we can simplify the equation to:

v1 = v1x + v2x

Substituting in the x components calculated earlier, we get:

v1 = v1*cos(30) + v2*cos(60)

Similarly, for the y components, we can set up the equation:

0 = v1y + v2y

Since the initial momentum is in the x direction, there is no initial momentum in the y direction. Substituting in the y components calculated earlier, we get:

0 = v1*sin(30) + v2*sin(60)

We now have two equations with two unknowns (v1 and v2), so we can solve for their values.

Solving the first equation for v1, we get:

v1 = (v2*cos(60) - v2*cos(30))/cos(30)

Solving the second equation for v2, we get:

v2 = -(v1*sin(30))/sin(60)

Substituting the first equation into the second, we get:

v2 = (-v2*cos(30)*sin(30))/(cos(30)*