# Kinematics: Two engines, One Rocket, and Displacement.

## Homework Statement

A Rocket with an engine that causes a net acceleration of 4m/s^2 launches with an initial velocity of 20m/s. The engine fuel lasts for 0.8s. A secondary engine then fires causing a net acceleration of 2m/s^2 for 0.6s How far off the ground is the rocket when the secondary rocket runs out of fuel?

## Homework Equations

• Vf=Vi+a⋅t
• Δd=Vi⋅t+2a⋅Δd
• Δd=1/2(Vf+Vi)t
(Vf: Final Velocity, a= Acceleration, t= Time, Δd= Displacement)

## The Attempt at a Solution

1. First I drew a picture of what this would look like including the data.
2. I listed the data.

Given:
Engine 1(Points A-B)[/B]
a: 4m/s^2
Vi: 20m/s
t: 0.8s

Engine 2(Points B-C)
a:2m/s^2
t: 0.6s

3. I'm assuming that each engine has their own separate equation. I'm thinking that displacement is what needs to be found for each engine at their own points and then added to find the total?

4. For engine 1 I thought that the equation for the final velocity should be used since I guess that is needed for the second engines initial velocity.

Vf=Vi+a⋅t
I substituted values and got 23.2m as the final velocity

~Im not sure what to do next or even if that was the correct step to first take.

kuruman
Homework Helper
Gold Member
2021 Award
Drawing pictures and listing the equations are good starting points, but what is your strategy? What is your plan of attack and how are you going to implement these equations? Note that the rocket motion has three separate time intervals and three separate constant accelerations during these intervals until it reaches maximum height.

Yes, you are so posed to solve for the final acceleration after the first stage and you are correct in saying that the final velocity of the first stage should be 23.2m/s,
I'm assuming that each engine has their own separate equation. I'm thinking that displacement is what needs to be found for each engine at their own points and then added to find the total?
this is also correct so what I would suggest is using the formula that factors in all your variables and solves your displacement as you had suggested, once you do that feel free to reply with what you came up with and I will tell you if that is what I also got.

Yes, you are so posed to solve for the final acceleration after the first stage and you are correct in saying that the final velocity of the first stage should be 23.2m/s,
this is also correct so what I would suggest is using the formula that factors in all your variables and solves your displacement as you had suggested, once you do that feel free to reply with what you came up with and I will tell you if that is what I also got.

I got 17.25m as the displacement for the first engine using this equation: Δd=Vi⋅t+2a⋅Δd
so is the final velocity for the first equation the initial for the second engine?

I got 17.25m as the displacement for the first engine using this equation: Δd=Vi⋅t+2a⋅Δd
so is the final velocity for the first equation the initial for the second engine?
Yes, the initial velocity for the second stage is the same as the final velocity for the first. Also I got 17.28 for the displacement of the first stage solved using x = x₀ + v₀t + ½at² = 0 + (20)(.8) + (1/2)(4)(.8)^2 (only the .8 is squared) so I don't know if your teacher considers the .03 negligible or not

Yes, the initial velocity for the second stage is the same as the final velocity for the first. Also I got 17.28 for the displacement of the first stage solved using x = x₀ + v₀t + ½at² = 0 + (20)(.8) + (1/2)(4)(.8)^2 (only the .8 is squared) so I don't know if your teacher considers the .03 negligible or not

Well, she doesn't really teach. It's hard teaching myself. That's why I'm here. So then I used this equation: Vf^2= Vi^2+2a⋅Δd to try and solve for the displacement for engine 2. and I got 13.1175m. I added that to the first displacement and got 30.38m. but the question states that the answer is 31.6m am I missing something? or am I not finished?

Well, she doesn't really teach. It's hard teaching myself. That's why I'm here. So then I used this equation: Vf^2= Vi^2+2a⋅Δd to try and solve for the displacement for engine 2. and I got 13.1175m. I added that to the first displacement and got 30.38m. but the question states that the answer is 31.6m am I missing something? or am I not finished?

that was after i solved for the final velocity of the second engine. if that was what i was supposed to do.

Well, she doesn't really teach. It's hard teaching myself. That's why I'm here. So then I used this equation: Vf^2= Vi^2+2a⋅Δd to try and solve for the displacement for engine 2. and I got 13.1175m. I added that to the first displacement and got 30.38m. but the question states that the answer is 31.6m am I missing something? or am I not finished?
Your process is correct however I am assuming you made a mistake in your math while solving through the exact process you described I got 31.6, what were the values you used for the initial velocity for the second stage and the final velocity for the second stage?

So i just used a different equation. Δd=Vi⋅t+1/2a⋅t . To find the displacement for the second engine. Vi=23.2m/s, t=0.6s, a=2m/s^2. I got 14.21m. then i added that to the first displacement and got 31.5m. not 31.6m

So i just used a different equation. Δd=Vi⋅t+1/2a⋅t . To find the displacement for the second engine. Vi=23.2m/s, t=0.6s, a=2m/s^2. I got 14.21m. then i added that to the first displacement and got 31.5m. not 31.6m
Probably the result of the first step getting a value of 17.25 instead of the 17.28 that I got, the formula you used for the first stage's displacement I've never seen before and when I solve completely for the problem I reach 31.56 which rounds to 31.6. so maybe there was an error in your math for the first stage or the formula is just wack, either way, I prefer the one I used to solve as it's used on the AP exams.

Probably the result of the first step getting a value of 17.25 instead of the 17.28 that I got, the formula you used for the first stage's displacement I've never seen before and when I solve completely for the problem I reach 31.56 which rounds to 31.6. so maybe there was an error in your math for the first stage or the formula is just wack, either way, I prefer the one I used to solve as it's used on the AP exams.

i redid that calculation and i did get 17.28 this time. I guess it was the math. Thanks for the help

i redid that calculation and i did get 17.28 this time. I guess it was the math. Thanks for the help
Great and anytime, also keep up the work teaching yourself the material I currently am having to do something similar as my school wouldn’t let me in the AP Physics C E&M course.