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
Please helppp <3
To determine the magnitude of v1 and v2, we can apply the conservation of momentum. According to this principle, the total momentum before the explosion is equal to the total momentum after the explosion.
The momentum of an object is defined as the product of its mass and velocity. In this case, since the two pieces have equal mass, we can assume the mass of each piece is m.
Before the explosion:
The total momentum before the explosion is the momentum of the complete rocket, which can be calculated using the formula P = m * v, where m is the mass and v is the velocity. Therefore, the momentum before the explosion is P_before = m * v_before.
After the explosion:
We can break down the velocity v1 into its horizontal and vertical components using trigonometry. Given that v1 forms an angle of 30 degrees in the right triangle, the magnitude of its horizontal component, v1x, is given by v1x = v1 * cos(30°), and the magnitude of its vertical component, v1y, is given by v1y = v1 * sin(30°).
Similarly, for v2 forming an angle of 60 degrees, the magnitude of its horizontal component, v2x, is given by v2x = v2 * cos(60°), and the magnitude of its vertical component, v2y, is given by v2y = v2 * sin(60°).
The total momentum after the explosion can be calculated by summing the momenta in the x and y directions. Therefore, P_after = (m * v1x + m * v2x, m * v1y + m * v2y).
Using the principle of conservation of momentum, we equate P_before to P_after:
m * v_before = m * v1x + m * v2x, m * v1y + m * v2y.
Cancelling out the mass term from both sides gives:
v_before = v1x + v2x, v1y + v2y.
Now, we have two equations:
1. v_before = v1x + v2x,
2. 0 = v1y + v2y.
To solve these equations, we substitute the expressions for the components discovered earlier.
1. v_before = v1 * cos(30°) + v2 * cos(60°),
2. 0 = v1 * sin(30°) + v2 * sin(60°).
Substituting the given value of v_before and solving the system of equations will provide the magnitude of v1 and v2.
Let's calculate the values step-by-step.
Given:
v_before = 48.4 m/s
v1 forms a 30° angle in the right triangle.
v2 forms a 60° angle in the right triangle.
Step 1: Calculate the components of v1 and v2.
v1x = v1 * cos(30°)
v1y = v1 * sin(30°)
v2x = v2 * cos(60°)
v2y = v2 * sin(60°)
Step 2: Equate v_before to the sum of the components.
v_before = v1x + v2x
0 = v1y + v2y
Step 3: Substitute and solve the equations.
48.4 = v1 * cos(30°) + v2 * cos(60°)
0 = v1 * sin(30°) + v2 * sin(60°)
You can now solve these equations using a calculator or numerical methods to find the values of v1 and v2.
To solve this problem, we can use the principle of conservation of momentum, which states that the total momentum before an event is equal to the total momentum after the event.
1. Step: First, let's calculate the initial momentum of the fireworks rocket.
Initial momentum (before the breakup) = mass × velocity
The mass of the fireworks rocket is not given, so let's assume it is "m".
Initial momentum = m × 48.4 m/s
2. Step: Next, let's consider the two pieces of the rocket after the breakup.
The triangle formed by v1, v2, and the initial velocity can be considered as a right triangle.
Using trigonometric ratios, we can determine the magnitudes of v1 and v2.
(a) Magnitude of v1:
In the right triangle, the angle between the initial velocity and v1 is 30 degrees.
Using the trigonometric ratio cosine, we can find v1:
cos(30°) = adjacent/hypotenuse
cos(30°) = v1/48.4 m/s
Solving for v1:
v1 = cos(30°) × 48.4 m/s
(b) Magnitude of v2:
In the right triangle, the angle between the initial velocity and v2 is 60 degrees.
Using the trigonometric ratio cosine, we can find v2:
cos(60°) = adjacent/hypotenuse
cos(60°) = v2/48.4 m/s
Solving for v2:
v2 = cos(60°) × 48.4 m/s
After following these steps and substituting the values, you should be able to find the magnitudes of v1 and v2.