A model rocket is fired vertically upward from rest. Its...

A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is $ a(t) = 60t $, at which time the fuel is exhausted and it becomes a freely "falling" body. Fourteen seconds later, the rocket's parachute opens, and the (downward) velocity slows linearly to $ -18\;ft/s $ in 5 seconds. The rocket then "floats" to the ground at that rate. (a) Determine the position function $ s $ and the velocity function $ v $ (for all times $ t $). Sketch the graphs of $ s $ and $ v $. (b) At what time does the rocket reach its maximum height, and what is that height? (c) At what times does the rocket land?

A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is $ a(t) = 60t $, at which time the fuel is exhausted and it becomes a freely "falling" body. Fourteen seconds later, the rocket's parachute opens, and the (downward) velocity slows linearly to $ -18\;ft/s $ in 5 seconds. The rocket then "floats" to the ground at that rate.
(a) Determine the position function $ s $ and the velocity function $ v $ (for all times $ t $). Sketch the graphs of $ s $ and $ v $.
(b) At what time does the rocket reach its maximum height, and what is that height?
(c) At what times does the rocket land?

Key Concepts

Determining Maximum Height in Projectile Motion

Determining the maximum height in a projectile motion problem typically involves finding the point in time when the object's upward velocity becomes zero. This condition is derived by setting the velocity function equal to zero and solving for time, which then allows one to compute the corresponding position or height.

Graphical Analysis of Motion

Graphical analysis of motion includes sketching the velocity and displacement functions to visually interpret the behavior of the moving object over time. Graphs help illustrate periods of acceleration, deceleration, and constant velocity, and they aid in understanding key events like reaching maximum height or landing.

Initial Value Problems

Initial value problems involve finding solutions to differential equations that satisfy given initial conditions. In the context of motion, these conditions, such as initial velocity or initial position, are used to determine the constants of integration when constructing the velocity and position functions from the known acceleration.

Piecewise Functions

Piecewise functions are used to define different expressions for different intervals of the independent variable. In many motion problems, forces such as thrust, free-fall, or aerodynamic deceleration act over distinct time periods, creating different regimes of acceleration. This concept is crucial for modeling realistic motions where behavior changes abruptly at certain moments.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without necessarily considering the forces causing the motion. In problems involving acceleration, velocity, and displacement, kinematics provides the fundamental relationships through which one can analyze the motion of an object over time using calculus.

Integration and Differentiation in Motion Analysis

Integration and differentiation are central mathematical tools in motion analysis, allowing one to link acceleration, velocity, and position. By differentiating the position function one gets velocity, and by differentiating velocity one gets acceleration. Conversely, integrating acceleration yields velocity and integrating velocity yields position, with constants of integration determined from initial conditions.

Transcript

00:01 The problem is, a model rocket is fired vertically upward from rust.

00:07 Its acceleration from the first three seconds is a t is equal to 60 times t, at which time the field is exhausted and it becomes a freely falling body.

00:23 14 seconds later, rocket protrude opens and the downward.

00:31 Velocity slows linearly to negative 18 feet per second in 5 seconds.

00:41 The rocket then floats to the ground at that rate.

00:48 Part a determine the position function and the velocity function.

00:53 Sketch the graphs of s and v.

01:00 The first one, t is between 0 and 3.

01:05 We have 80 is equal to 60 times t then we have weighty is equal to 30 times t squared and s t is equal to 10 times t's cube and when t is equal to three we have v3 is equal to 200 and 70 s 3 is also 270.

02:06 When t is between 3 and 17, so we have weighty is equal to negative 32 is this is negative g times t minus 3 plus c constant number c and when t is equal to 3, we have v3 is equal to 270.

02:47 So c is equal to 270.

02:54 And the g is equal to 32.

02:56 So this is negative 32 times t minus 3 plus 270.

03:06 St is equal to the negative 32 times 1 half times t minus 3 square plus 270 times t minus 3 and plus c and when t is equal to 3 we have st is equal to 270 so here c is also 270, so this is equal to negative 16 times t minus 3 square plus 270...

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