A fireworks rocket is moving at a speed of 48.4 m/s. The rocket suddenly breaks into two pieces of equal mass, which fly off with velocities v1 and v2, as shown in the drawing. What is the magnitude of (a) v1 and (b) v2?

In the image the two pieces are going off in a right traingle.
V1 having an angle of 30 degrees and
V2 having an angle of 60 degrees

To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before the explosion is equal to the total momentum after the explosion.

Let's denote the mass of each piece of the rocket as "m" and the magnitude of v1 and v2 as v.

The total momentum before the explosion is given by:
P(before) = m * 48.4 m/s

After the explosion, the two pieces of the rocket move in different directions. To find the magnitude of v1 and v2, we can use trigonometry.

Using the trigonometric properties of a right triangle, we can write:
v1 = v * cos(30°)
v2 = v * sin(60°)

Now, let's substitute the values into the conservation of momentum equation and solve for v:
P(before) = P(after)
m * 48.4 m/s = (m * v1) + (m * v2)
48.4 = v * cos(30°) + v * sin(60°)

To solve for v:
48.4 = v * cos(30°) + v * √3/2
48.4 = v * (√3/2 + 1/2)
v = 48.4 / (√3/2 + 1/2)

Now we can calculate the magnitude of v1 and v2 using the given angles:
(a) To find v1:
v1 = v * cos(30°)
v1 = (48.4 / (√3/2 + 1/2)) * cos(30°)

(b) To find v2:
v2 = v * sin(60°)
v2 = (48.4 / (√3/2 + 1/2)) * sin(60°)

Calculating these values will give you the magnitude of v1 and v2.

To find the magnitudes of v1 and v2, we can use the physics concept of conservation of momentum. According to this principle, the total momentum before and after the explosion should be the same.

Let's denote the initial mass of the fireworks rocket as m, and each of the two pieces after the explosion as m1 and m2.

The total momentum before the explosion can be calculated as:
P_initial = m * v_initial

The total momentum after the explosion can be calculated as:
P_final = m1 * v1 + m2 * v2

Since the two pieces have equal mass, we can assume m1 = m2 = m/2.

Now, let's calculate the magnitudes of v1 and v2:

(a) To find the magnitude of v1, we can use the given angle of 30 degrees. We can use trigonometry to find the horizontal component of velocity, and multiply it by the magnitude of the velocity.

v1_x = v1 * cos(30 degrees)

Since the triangle is a right triangle, the vertical component of v1 will be the same as the magnitude of v1 times sin(30 degrees).

v1_y = v1 * sin(30 degrees)

To find the magnitude of v1, we can use the Pythagorean theorem:

v1^2 = v1_x^2 + v1_y^2

Once you have the value of v1, calculate its square root to get the magnitude.

(b) To find the magnitude of v2, we can use the given angle of 60 degrees. Using the same logic as above, we can determine the horizontal and vertical components of v2.

v2_x = v2 * cos(60 degrees)
v2_y = v2 * sin(60 degrees)

To find the magnitude of v2, we can again use the Pythagorean theorem:

v2^2 = v2_x^2 + v2_y^2

Once you have the value of v2, calculate its square root to get the magnitude.

In summary:
(a) Calculate v1_x = v1 * cos(30 degrees) and v1_y = v1 * sin(30 degrees). Then, use v1^2 = v1_x^2 + v1_y^2 to find the magnitude of v1.
(b) Calculate v2_x = v2 * cos(60 degrees) and v2_y = v2 * sin(60 degrees). Then, use v2^2 = v2_x^2 + v2_y^2 to find the magnitude of v2.