When it is 145 m above the ground, a rocket traveling...

When it is 145 $\mathrm{m}$ above the ground, a rocket traveling vertically upward at a constant 8.50 $\mathrm{m} / \mathrm{s}$ relative to the ground launches a secondary rocket at a speed of 12.0 $\mathrm{m} / \mathrm{s}$ at an angle of $53.0^{\circ}$ above the horizontal, both quantities being measured by an astronaut sitting in the rocket. After it is launched the secondary rocket is in free-fall. (a) Just as the secondary rocket is launched, what are the horizontal and vertical components of its velocity relative to (i) the astronaut sitting in the rocket and (ii) Mission Control on the ground? (b) Find the initial speed and launch angle of the secondary rocket as measured by Mission Control. (c) What maximum height above the ground does the secondary rocket reach?

Key Concepts

Kinematic Equations in Free Fall

Kinematic equations provide relationships between displacement, initial velocity, time, and acceleration for objects moving under constant acceleration, such as free-fall due to gravity. These equations are essential for calculating quantities like maximum height, time of flight, and final velocity in projectile motion problems.

Projectile Motion

Projectile motion describes the trajectory of an object that is launched into the air and is subject only to gravitational acceleration (ignoring air resistance). It involves independently analyzing the horizontal constant velocity and the vertical acceleration due to gravity to determine key parameters such as maximum height and range.

Relative Motion

This concept involves understanding how an object's velocity and acceleration differ when observed from different reference frames. In problems like these, one must account for the addition or subtraction of velocities to convert measurements from a moving frame (e.g., the astronaut's frame) to a stationary frame (e.g., Mission Control's frame).

Vector Decomposition

Vector decomposition is the process of breaking a given vector into its horizontal and vertical components using trigonometric functions. This simplification allows separate analysis of motion in the two mutually perpendicular directions, which is crucial in solving kinematics and projectile motion problems.

Transcript

00:01 In this question, we have a big rocket, okay? so initially rise up and then with a constant speed of 8 .5 meters per second.

00:23 And then at a height of 145 meters, it launches a second rocket.

00:34 That is an angle of 53 degrees on the horizontal.

00:42 Okay.

00:44 Then the speed is 12 meters per second.

00:50 But in this thing, this information is in the astronaut's frame.

01:05 Not in the ground frame.

01:09 So we need to find the horizontal.

01:12 Horizontal and vertical components of the secondary rocket, in the astronaut's frame and in the ground frame.

01:22 We also need to find the initial speed and launch angle of the secondary rocket measured by the ground frame and maximum height reached by the secondary rocket.

01:34 So again, this is a problem on relative motion and projectile motion.

01:41 So to solve this problem, okay, in part a, okay, and part i in the astronaut's frame, okay, so the velocity of rocket in the astronaut's frame frame, the x component is 12 cosine degrees, calculate this you get 7 .22, it does a second, and then the y component of the velocity of the rocket and the astronaut's frame is 12 sine 53 degrees and you get 9 .5 .8 meters per second.

02:33 So this is the answer for ai and then for a.

02:41 A.

02:41 In the ground frame or the mission, yeah, i'll just take it to be the ground frame.

02:52 Okay.

02:53 So the velocity.

02:54 So the x component of the velocity of the rocket in the ground frame is using the velocity relative velocity equation.

03:07 So it is a velocity of the rocket in the astronaut's frame, the x component plus the velocity of the astronaut in the ground frame, the x component.

03:19 And when you do this, you get 7 .22 plus 0.

03:25 So this is 7 .22 meters per second.

03:29 Okay and for the y component, okay, just do the same thing is that now you change x to y...

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