A 7500-kg rocket blasts off vertically from the launch pad with a constant upward acceleration o

A 7500-kg rocket blasts off vertically from the launch pad...

Key Concepts

Constant Acceleration

This concept refers to situations where an object's acceleration remains unchanging over time. It is fundamental in kinematics because it allows the use of standard equations to predict parameters like displacement, velocity, and time from known values. When acceleration is constant, the motion can be accurately described and analyzed, simplifying the calculation of trajectories and maximum displacements.

Projectile Motion

Projectile motion describes the behavior of an object that is launched into the air and moves under the influence of gravity alone after its engines or propulsion stop. This motion is characterized by an initial phase while powered and a subsequent phase where only gravity acts. Understanding projectile motion is key in analyzing how objects ascend, decelerate, reverse direction at the peak, and accelerate as they fall back under gravitational pull.

Kinematics Equations

The kinematics equations are mathematical formulas that relate displacement, initial velocity, final velocity, acceleration, and time during motion. They are essential tools in solving problems involving both constant acceleration and free-fall conditions. By applying these equations, one can determine quantities such as the maximum height reached after engine failure, the duration of the flight, or the impact speed on landing, independent of the mass of the object.

Graphical Analysis of Motion

Graphical analysis involves sketching graphs like acceleration versus time, velocity versus time, and displacement versus time to visually represent an object's motion. These graphs provide intuitive insights into how the object’s speed and position change over time and allow for easier interpretation of acceleration phases, deceleration under gravity, and overall motion behavior. They help in identifying key points such as peak height and the moments of engine failure or impact.

Transcript

00:01 The first step to doing this problem is to figure out what is actually happening.

00:05 Originally, we have a rocket on the ground floor, and it's boosted by constant acceleration of 2 .25 meters per second squared, until it reaches a point where the engines break at 525 meters up.

00:16 And then it's only affected by the acceleration due to gravity, negative 9 .8, until it reaches a maximum height, and then it drifts back down to the ocean ground, or to the earth ground.

00:27 Now, the first step to solving part a is to figure out, what velocity it is at at this level here.

00:36 To do that, we use kinematic equation 3 here because we know that v0 is equal to 0.

00:42 So v squared is equal to 2a x minus x0.

00:47 We know a is 2 .25 meters per second squared and x minus x not is 525 meters.

00:53 Plugging that in, we get that v is equal to 48 .6 meters per second.

01:03 Now that we know the velocity at the point where the engines break, we need to figure out the time it takes for the rocket to get to its maximum height.

01:11 To do that, we use kinematic equation 1, which is v equals v0 plus a times t.

01:18 In this case, a is equal to negative 9 .8.

01:21 T is unknown, and then v minus v .0 is zero for v, because at the maximum height, v will be zero.

01:29 And then v .0 is actually this value.

01:32 And so we get negative 48 .6 is equal to negative 9 .8 times t.

01:38 So t is equal to 4 .96 seconds.

01:47 Now to get the x value here, we can use kinematic equation 2.

01:54 So x minus x not, where x not is 525 meters, is equal to v0, and in this case, v0 is 48 .6 times t 4 .96 plus 1⁄2 negative 9 .8, 4 .96 squared.

02:23 Solving this for x, we get that x is equal to 120 .51 meters.

02:29 That is the x in between the maximum height and when the engines break.

02:33 So it's not the maximum height.

02:35 To get the maximum height, we have to take this value and add it to the 525.

02:44 And so we get 645 .51 for the maximum height.

02:49 And that's the answer to part a.

02:54 For part b, we need to figure out the time it takes for the rocket to reach all the way back down to the ground.

02:59 And the velocity that it is at that point.

03:05 So let's look at this a little further.

03:08 We know that the time it takes the rocket to get from here to the x is 4 .96 seconds, which also means it takes 4 .96 seconds for it to fall back down to this height here.

03:26 We also need the time it took the rocket to get from the ground floor initially up to a height of 525.

03:33 And so for this, we can use kinematic equation 1.

03:39 And so this time here, the final velocity is 48 .6, and it's equal to 2 .25, because that's the acceleration, times t, that yields a t value of 21 .6 seconds.

04:01 And so it takes 21 .6 seconds for the rocket to get initially too high to 525 meters.

04:06 It takes 4 .96 to go up until it's at rest, and then 4 .96 to go back down.

04:13 And so lastly, we need the time it takes to get from 525 down to zero again.

04:25 And so it's actually easiest to calculate the final velocity at this point.

04:32 And then we can use that to find the time it takes to get to that point.

04:36 To find the final velocity at this point, we're going to use kinematic equation three again.

04:41 And so i'm going to go to a new sheet here.

04:47 And using kinematic equation three, i get that v squared is equal to, 2 ,362 plus 10 ,290.

04:56 And i got that from using this.

04:59 V -0 is negative 48 .6 at this point after it falls down.

05:06 A is still negative 9 .8.

05:08 And then x minus x -nod is negative 525.

05:12 And then whenever you plug those numbers, and then you get these.

05:15 And then you add this together and take the square root, and you get that v is equal to negative 112 .5.

05:23 When you take the positive square root, you won't get a negative here.

05:25 But we know that since the rocket is moving down, it must have a negative sign.

05:29 So we can take the negative square root of this.

05:31 And that's what we should do.

05:33 This is a meters per second, by the way...

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