Quick Projectile Motion Problem

[quicksilver123]

Homework Statement

A child sitting in a tree throws his apple core from where he is perched (a height of 4.0m) with a velocity of 5.0m/s [35 deg above horizontal] and it hits the ground right next to his friend.

a)how long is it before the apple core hits the ground?
b)how far from the base of the tree will the apple core land?
c)what is the velocity of the apple core on impact?

Homework Equations

suvat
pythag.
quad. form.

The Attempt at a Solution

Given:
V_i=5.0m/s [35 deg above horizontal]
Δd_y = 4.0m

a)

Find Δt.

Let [down] and [forward] be positive.

Initial Velocity Components:
V_ix = 5m/s (cos35) = 4.095760221m/s
V_iy = 5m/s (sin35) = -2.867882182m/s
*Note: I threw the negative on the y-component because it is traveling upwards.

Acceleration Components:
a_x = 0m/s/s
a_y = 9.8m/s/s

Calculation for part A:
Δd_y = V_iyΔt +1/2 (a_y)Δt²
4.0m = -2.867882182m/s Δt + 4.9m/s/s (Δt²)

In order to isolate the time variable, I rearranged to the form ax²+bx+c=0 and solved via the quadratic formula.
The only real root yielded was x=1.242359580605109

Therefore, Δt = 1.242359580605109 seconds


b)
Find Δd_x.

Calculation for part B:
Δd_x = V_ixΔt +1/2 (a_x)Δt²
Δd_x = 4.095760221m/s (1.242359580605109 seconds) + 0
Δd_x = 5.08840695042 m


c)
Find V_f

Components of V_f:
V_fx
V_fx² = V_ix²+2a_xΔd_x
Since a_x=0m/s/s, V_fx=V_ix
∴ V_fx=5m/s

V_fy
V_fy² = V_iy²+2a_yΔd_y
V_fy² = 8.22474821m²/s² + 78.4m²/s²
V_fy = √86.62474821
V_fy = 9.307241708m/s

V_f = √(V_fx² +V_fy²
V_f = √14.30724171
V_f = 3.782491469m/s



Answers rounded:

Δt = 1.24 seconds
Δd_x = 5.088 m
V_f = 3.78 m/s

I know the sig digits aren't correct but I didn't want to sacrifice accuracy.


Are these answers correct?

 

[Simon Bridge]

"Viy = 5m/s (sin35) = -2.867882182m/s"

missed a minus sign from in front of the 5 :)

Note: you are being trained to be able to solve problems where nobody knows the answer.
This means you need to figure out how to tell if you've got it right without having to ask someone else. Problems like the one you just did are great for practicing this - for instance,
Does the result you got make sense?
Can you identify the source of your uncertainty with your answers?
Is there another way of doing the problem, eg. graphically, that will help you tell if it's right or not?

 

[quicksilver123]

I'm not really sure how to tell if I got it right.
Obviously there are some cases where a wrong answer will make absolutely no sense (eg. in this case - Δd_x = -500m would be blatantly incorrect) but the majority of errors that go uncaught will be more subtle. You seem to have some ideas on this topic, could you share them?
I like your point about finding points of uncertainty in calculations.
Doing the problem graphically, at least in this case (I think), would only help with a visual representation. I guess that one could use calculus to find the point where the projectile reached the maximum height, and then calculate the downward trajectory from there in the context of the problem to get an answer... but that doesn't seem like the most straightforward way to attack something like this.

 

[Simon Bridge]

One of the ways to gain confidence in your answer is to do the algebra first - then add the numbers in.
That way you keep track of the relationships between the different quantities. Once you put the numbers in the relationships are less obvious.

If you try the graphical method - you draw the two v-t diagrams. One for Vx and one for Vy. It's a line from (t,v(t)) = (0,-Vi) to (T,+Vf) for each component so it crosses the t axis making two triangles for Vy (and a box for Vx) - you know the area of the second triangle - it's given - and you know the slope of the line. Draw them one above the other and make their time axis coincide - so they both finish at the same place for T.

Your equations will end up consistent with the diagrams which helps you be confident about them as opposed to just using memorized formulae. I figured you'd like this approach because you avoided the projectile equations - which most people would use - favoring your native understanding. Most people would stick to using "up" as positive too. This is good - you'll understand more and it is ultimately less work.

The real trick is to identify where you feel uncertain/unsure about the process - which bits feel intuitively "iffy"? Those are the bits you need to check.

For reasonableness of your answer - you realize that $v_f$ is actually the magnitude of a vector - is that vector pointing in the right direction? Is the magnitude consistent with the components? i.e.

Vfx=5m/s
Vfy = 9.3m/s
Vf = 3.78 m/s

... so Vfx is exactly the same as the total initial velocity - does this seem right?

... Vfy is about twice Vfx - would that be about right? If the apple-core was just dropped from the branch instead of thrown, how fast would it go when it hit? Is that consistent with what you got?

... Vf is the hypotenuse of a right-angled triangle, yet it is smaller that both the other sides... is this right?

... see what I'm talking about?
When you get an answer - think about what it means and how it relates to other things you know.

 

[Nickie]

I would say that your calculations and approach seem correct and the answers are reasonable. However, it is always important to double-check your work and ensure that you are using the correct equations and units throughout your calculations. Additionally, make sure to use the correct number of significant digits in your final answers. Overall, great job on solving this projectile motion problem!