How Does an Airplane's Climb Affect Package Drop Distance?

Click For Summary

Homework Help Overview

The problem involves an airplane climbing at a specified speed and angle, releasing a package at a certain altitude. The discussion focuses on calculating the horizontal distance the package travels before hitting the ground and determining the angle of its velocity vector just before impact.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss breaking down the problem into velocity components and using kinematic equations. There are attempts to calculate the time of flight and the horizontal distance traveled by the package.

Discussion Status

Participants are actively engaging with the problem, exploring different approaches to find the time it takes for the package to fall and how to calculate the horizontal distance. Some guidance has been provided regarding the use of kinematic equations and the importance of considering the initial conditions of the package upon release.

Contextual Notes

There are discussions around the assumptions made regarding the direction of acceleration due to gravity and the interpretation of time values obtained from calculations. Participants are also reflecting on the implications of their results in the context of the problem.

kcalhoun
Messages
22
Reaction score
0

Homework Statement



An airplane with a speed of 59.6 m/s is climbing upward at an angle of 40° counterclockwise from the positive x axis. When the plane's altitude is 600 m the pilot releases a package.
(a) Calculate the distance along the ground, measured from a point directly beneath the point of release, to where the package hits the earth.

(b) Determine the angle of the velocity vector of the package just before impact.

Homework Equations



Eight kinematics equations

The Attempt at a Solution



completely stuck.. i don't even know where to begin
 
Physics news on Phys.org
Welcome to PF kcalhoun!

This question requires you to split the problem into velocity components and then use your constant acceleration formulae. Also think about what the package does once it is released. So to start with, can you tell us what the speed is of the plane in the horizontal (positive x) and vertical directions (positive y)?

The Bob
 
for the horizontal it would be 59.6cos40
and the vertical would be 59.6sin40
 
Nice. So, if we can find out how long the package takes to fall to the ground, then it would seem logical that we can find out how far it travels due to time. So what do we now need to consider?

The Bob
 
solve for t in one of the constant acceleration equations?
 
Yes, in short. However, remember to consider the conditions the package is under once released. What is it's inital vertical velocity? etc. The key is what is positive and what is negative.

The Bob
 
ok so using the equation vy=v0y+ayt i get.. 0=59.6sin40+(-9.80)t and t = 3.909?
 
That wasn't the one I was thinking of. What you've worked out is the time it takes for the package to come to rest whilst in mid-air.

The Bob
 
I'm not sure what you are looking for then...
 
  • #10
Okay. Once the package is released, it continues to travel upwards as it still has a vertical component. Gravity takes effect and slows it down such that its velocity after 3.909s is 0ms-1. So, you can either start over with one equation that will do it all for you (e.g. to get total time) or you can stick with this and you can now work out how far it has traveled vertical in addition to the 600m and thus the total time for it to be released and fall to the ground.

Do you understand why what you've worked out is the time for it to be at v = 0 in mid-air?

The Bob
 
  • #11
not really no but, was this the equation that you were thinking of... y=v0y+1/2ayt^2?
 
  • #12
That was the one I was thinking of but I'd rather you understood why. The plane ascends and the package is released. The package has the same vertical component as the plane and it is immediately constant, e.g. no external force, after released. However, gravity now acts on the package. (At this point I should note that I will assume negative y is actually positive and positive y is negative but it doesn't really matter that much.) So the package is traveling at -59.6sin(40). This is our initial velocity. We also know at what height above the ground the package was released (600m) and we know that a = g = 9.8ms-2. Does that help you see what I'm getting at? Also, the reason t = 3.909s is write for what I said is because the formula automatically works out the next time something happens (in this case when v = 0).

The Bob
 
  • #13
ok yeah that makes sense so now when i solve for t i get 114.631 then i need to find delta x to solve part a?
 
  • #14
The equation is:
s = u*t + 0.5*a*t^2
where s is the displacement, u is the intial velocity, t is time and a is the constant acceleration. Your equation in post 11 had a missing t, my bad... sorry. I'm not doing very well here.

The Bob
 
  • #15
right i used that equation to solve for t and i got 600=59.6sin40t+1/2(-9.8)t^2 which yields 114.631 right? then where do i go from there?
 
  • #16
Can you put your working out up please? What you have will yield an imaginary number.

The Bob
 
  • #17
(600-59.6sin40)/(1/2*9.8)=114.631 does that look right?
 
  • #18
Where have your 't's gone? You can't solve for t without there being any in the equation. This is a quadratic equation so you're need the quadratic formula.

The Bob
 
  • #19
what i did was take the original equation y=v0yt+1/2ayt^2 and divided by t to get y=v0y+1/2ayt and the i solved for t to get t=(y-v0y)/(1/2ay) and then i plugged in the numbers.. t=(600-59.6sin40)/(1/2*9.8) so t=114.631 is that not right?
 
  • #20
It's a good idea but it will not work here because 't' could be positive or negative. You can only divide if you know it is always positive or always negative. So now try it as a quadratic.

The Bob

P.S. Sometimes it's worth thinking logically. It only takes an object, initially at rest, just over 11 seconds to fall 600 metres so 114.631 seconds should stand out as not being right.
 
  • #21
ok so after plugging the numbers into the quadratic equation i get 7.827s or -15.645s so t would bed 7.827s?
 
  • #22
Right, we know have the correct values but the signs are wrong. Do you not find that the value inside the square is negative?

The Bob
 
  • #23
so it should be -7.827 and 15.645?
 
  • #24
Yer but do you find a negative value in your square root? If so, then all we do is reverse the orientation, e.g. you had positive y as positive but now positive y direction is negative. Anyway, so we now have these two values. The positive value is positive time and the negative value is negative time, e.g. back in time, so obviously we take 15.6 seconds as the answer to how long does it take the package to fall to the ground.

(a) Calculate the distance along the ground, measured from a point directly beneath the point of release, to where the package hits the earth.

We now have the time it takes to fall so we can now apply this value to the horizontal component.

The Bob
 
  • #25
so use the equation x=1/2(v0x+vx)t and i get 714.29m sweet so now part b...?
 
  • #26
That is roughly what I get so we'll allow for errors in roundings. Do you understand how you got the answer though? There is no point explaining the next bit if this hasn't made any sense.

The Bob
 
  • #27
yeah i understand it that helped a lot thank you
 
  • #28
Cool, glad to here it.

kcalhoun said:
(b) Determine the angle of the velocity vector of the package just before impact.

How do you think we approach this bit then? What information do we know relating to the package just before it hits the ground?

The Bob
 
  • #29
its velocity is 59.6m/s
the x component is 59.6cos40
the y component is 59.6sin40
 
  • #30
The x component is right as we know it is constant but the y component changes with time because of the acceleration due to gravity.

The Bob
 

Similar threads

How Does an Airplane's Climb Angle Affect Package Trajectory on Release?
  • · Replies 1 ·
Replies
1
Views
4K
Physics Projectile Word Problem -- Dropping packages from an airplane
  • · Replies 20 ·
Replies
20
Views
5K
Find the initial velocity of the package relative to the ground
  • · Replies 18 ·
Replies
18
Views
2K
Projectile Motion problem with Plane dropping package
  • · Replies 2 ·
Replies
2
Views
8K
Why is my answer wrong? Projectile Motion
  • · Replies 3 ·
Replies
3
Views
4K
Airplane Dropping a Package Relative Motion
  • · Replies 2 ·
Replies
2
Views
3K
Tricky Kinematics: Projectile Motion in 2D
  • · Replies 3 ·
Replies
3
Views
2K
When Should an Airplane Drop a Package to Hit a Target?
  • · Replies 1 ·
Replies
1
Views
4K
Kinematics in two Dimension- projectile motion
  • · Replies 10 ·
Replies
10
Views
5K
What is the angle of the velocity vector of a released package?
  • · Replies 5 ·
Replies
5
Views
4K