A model rocket blasts off from the ground, rising straight...

A model rocket blasts off from the ground, rising straight upward with a constant acceleration that has a magnitude of 86.0 $\mathrm{m} / \mathrm{s}^{2}$ for 1.70 seconds, at which point its fuel abruptly runs out. Air resistance has no effect on its flight. What maximum altitude (above the ground) will the rocket reach?

Key Concepts

Segmented Motion Analysis

This analytical approach breaks the overall motion into distinct segments, such as an initial powered ascent with constant acceleration and a subsequent unpowered coasting phase affected only by gravity. Each segment is treated separately using the appropriate kinematic equations to determine characteristics like maximum altitude.

Gravity

Gravity is the force that attracts objects towards Earth, causing a uniform acceleration (approximately 9.81 m/s²). In the context of projectile motion, after propulsion stops, gravity acts as the sole acceleration affecting the object's motion, decelerating its upward movement until it reaches its peak height.

Constant Acceleration Motion

This concept involves analyzing objects experiencing a fixed, unchanging acceleration over a period of time. It is fundamental to understanding how velocity and displacement evolve when acceleration remains constant, and it utilizes specific equations that relate these quantities over time.

Kinematic Equations

These are formulas used in the study of motion that relate displacement, initial velocity, time, acceleration, and final velocity. They are essential tools in solving problems involving motion under constant acceleration, and they help determine various parameters of an object's trajectory.

Projectile Motion

This refers to the motion of an object that is launched into the air and moves under the influence of gravity alone after an initial phase of propulsion. It is typically divided into separate phases, such as powered flight and coasting upward, and is analyzed using kinematic methods.

Transcript

00:01 In this problem, we have a rocket that has two phases in its flight.

00:05 The first is when its engine is engaged, and it starts from rest so that initial velocity is zero meters per second, but it has a massive acceleration of 86 meters per second squared, which is about eight gs of acceleration.

00:21 But it only maintains this acceleration for 1 .7 seconds.

00:26 Now, this question is asking, after the engine cuts out, how high up will the rocket fly if we ignore air resistance.

00:34 So as soon as the engine cuts off, you might think the acceleration becomes zero.

00:38 Remember, as long as we're on the planet earth, that acceleration is going to be negative 9 .8 meters per second squared while it's in the sky.

00:46 And we're looking at how high up it will go, so that final velocity is going to be zero meters per second.

00:52 And this problem is looking for that maximum height, which is actually going to be what we get if we add our two x values together, because it's going to be traveling upwards during both phases here.

01:04 So the tissue that kind of ties these two parts together is that whatever speed we end at in phase one is what speed we start at in phase two.

01:12 So those two values there are going to be the same thing.

01:16 So that gives us a couple of goals, a couple of things to look for.

01:21 Let's first start by figuring out what that final velocity is going to be.

01:26 So in the first phase in phase one here, i have v .0 a .t, and i'm looking for final velocity, which means i can use that first equation on the list.

01:39 That final velocity is equal to v .0 plus a times t.

01:43 So all i need to do to get my final velocity here is multiply my acceleration and time together, which gives me an impressive 146 .2 meters per second.

01:55 So going pretty quickly by the time we reach the end of the engine.

02:01 And that value is both the final velocity of phase one.

02:04 And like we said at the start, it's the initial velocity for phase two...

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