A 9.3 kg firework is launched straight up and at its maximum height 45 m it explodes into three parts. Part A (0.5 kg) moves straight down and lands 0.29 seconds after the explosion. Part B (1 kg) moves horizontally to the right and lands 10 meters from Part A. Part C moves to the left at some angle. How far from Part A does Part C land (no direction needed)?
I got part C's initial velocity as 10.05m/s.
Now i want to use y = 10.05sinAt + 4.9t^2
to find t
then use t to find the distance. But how do i find the angle A part C moves at?
The angle that part C moves at can be figured out by applying a momentum balance.
Initial skyrocket momentum at maximum height = 0
= sum of the momenta of parts A, B and C
The mass of part C is 9.3 - 0.5 - 1.0 = 7.8 kg. Its upward momentum component is the same as part A's but in the opposite direction (up). Its horizontal momentum component is the same as part B's, but to the left.
I am neglecting the momentum of gases released in the separation explosion, but this is probably OK.
To find the angle at which Part C moves, you need to consider the horizontal and vertical components of its motion.
First, let's focus on the vertical motion of Part C. You correctly identified that the equation y = 10.05sin(At) + 4.9t^2 can be used to describe its vertical displacement. Here, y represents the height above the ground, t is the time since the explosion, and A is the angle at which Part C moves.
Since Part C moves to the left, the vertical component of its initial velocity can be considered negative. Therefore, the equation for the vertical velocity (Vy) can be written as Vy = -10.05sin(At).
Now, to find the angle A, we can use the fact that Part C lands at the same time as Part A. The equation for the vertical displacement of Part A can be written as y = -0.5*9.8t^2, where -0.5*9.8t^2 accounts for its downward motion.
We know that both Part C and Part A land at the same time, so we can equate the vertical displacements of Part C and Part A:
10.05sin(At) + 4.9t^2 = -0.5*9.8t^2
Simplifying the equation, we get:
10.05sin(At) + 4.9t^2 = -4.9t^2
Canceling out the t^2 terms, we have:
10.05sin(At) = 0
Since sin(At) = 0 for At = nπ, where n is an integer, we can conclude that At = 0 or At = π.
However, At = 0 corresponds to Part C not moving vertically, which is not possible in this scenario. Therefore, we have At = π.
Now that we know the angle A, we can proceed with finding t and the distance at which Part C lands using the equation you mentioned, y = 10.05sin(At) + 4.9t^2.