Question

A fireworks rocket is launched vertically into the night sky with an initial speed of 44.2

m/s. The rocket coasts after being launched, then explodes and breaks into two pieces
of equal mass 2.50 s later. (a) If each piece follows a trajectory that is initially at 45.0° to
the vertical, what was their speed immediately after the explosion? (b) What is the
velocity of the rocket’s center of mass before and after the explosion? (c) What is the
acceleration of the rocket’s center of mass before and after the explosion?

Answer 1

a)44.2 m/s

b)zero

c)zero

Step-by-step explanation:

(a) The initial vertical component of velocity is given by v0y = 44.2 m/s. At an angle of 45 degrees to the vertical, the initial horizontal and vertical components of velocity are equal, so each piece has an initial speed of v0 = v0y / sin(45) = 62.4 m/s. When the rocket explodes, the momentum is conserved in both the x- and y-directions, so the horizontal component of velocity remains the same while the vertical component changes sign. Therefore, each piece has a velocity of v = v0 / sqrt(2) = 44.2 m/s after the explosion.

(b) The velocity of the rocket's center of mass before the explosion is zero, since it is at rest. After the explosion, the two pieces move in opposite directions with equal and opposite momentum. Since they have the same mass, their velocities are equal in magnitude, and the velocity of the center of mass is zero.

(c) Before the explosion, the rocket is at rest, so its acceleration is zero. After the explosion, the two pieces have equal and opposite acceleration vectors,

(a) The initial vertical component of velocity is given by v0y = 44.2 m/s. At an angle of 45 degrees to the vertical, the initial horizontal and vertical components of velocity are equal, so each piece has an initial speed of v0 = v0y / sin(45) = 62.4 m/s. When the rocket explodes, the momentum is conserved in both the x- and y-directions, so the horizontal component of velocity remains the same while the vertical component changes sign. Therefore, each piece has a velocity of v = v0 / sqrt(2) = 44.2 m/s after the explosion.

(b) The velocity of the rocket's center of mass before the explosion is zero, since it is at rest. After the explosion, the two pieces move in opposite directions with equal and opposite momentum. Since they have the same mass, their velocities are equal in magnitude, and the velocity of the center of mass is zero.

(c) Before the explosion, the rocket is at rest, so its acceleration is zero. After the explosion, the two pieces have equal and opposite acceleration vectors, but since they have the same mass, their acceleration vectors cancel out and the acceleration of the center of mass is zero.