A rocket is fired vertically upward. At the
instant it reaches an altitude of 2300 m and a
speed of 279 m/s, it explodes into three equal
fragments. One fragment continues to move
upward with a speed of 224 m/s following the
explosion. The second fragment has a speed
of 358 m/s and is moving east right after the
explosion.
What is the magnitude of the velocity of
the third fragment?
Answer in units of m/s.
initial momentum up = (m1 + m2 +m3) (279) = Pi
initial momentum horizontal = 0
Final momentum up = m1 *224 + m3 v3up = Pi
Final momentum east = m2 *358 + m3 * v3e
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now all those masses are the same. I am going to call them each 1 kg
3 (279) = 1*224 + 1 * v3up
well, v3 up = 3 *279 - 224
and
v3 east = -358
To find the magnitude of the velocity of the third fragment, we need to break down the velocity into its horizontal and vertical components.
Let's consider the initial velocity of the rocket before the explosion. Since it was fired vertically upward, the vertical component of the initial velocity is positive, and the horizontal component is zero.
After the explosion, the first fragment has a velocity of 224 m/s directed vertically upward, so its vertical component is positive (+224 m/s), and its horizontal component is zero.
The second fragment has a velocity of 358 m/s and is moving east, so its horizontal component is positive (+358 m/s), and its vertical component is zero.
Since the explosion occurs at an instant, the total momentum before and after the explosion must be conserved.
Given that the rocket exploded into three equal fragments, the combined vertical components of all three fragments after the explosion must add up to zero to conserve the total momentum in the vertical direction.
So, the vertical component of the third fragment's velocity can be found by subtracting the sum of the vertical components of the first and second fragments from zero:
0 - (+224 m/s) - 0 = -224 m/s
Since the second fragment is moving east, the horizontal component of the third fragment's velocity should be the same as the second fragment's horizontal velocity:
Horizontal component of the third fragment's velocity = +358 m/s.
Now, we have the vertical and horizontal components of the third fragment's velocity (-224 m/s and +358 m/s, respectively). To find the magnitude of the velocity, we can use the Pythagorean theorem:
Magnitude of the velocity = √((vertical component)^2 + (horizontal component)^2)
= √((-224 m/s)^2 + (358 m/s)^2)
= √(50176 + 128164)
= √(178340)
≈ 422.35 m/s.
Therefore, the magnitude of the velocity of the third fragment is approximately 422.35 m/s.