# Imagine that a bullet is shot vertically into the air with an initial

## Homework Statement

Imagine that a bullet is shot vertically into the air with an initial speed of 9800 m/s. If we ignore air friction, how high will it go?

vi = 9800 m/s
h = ?

## Homework Equations

v2 = vi2 + 2aΔy
Ki = Ui --> 1/2mv2 = mgh

F = (Gm1m2)/r

## The Attempt at a Solution

Attempt 1

v2 = vi2 + 2aΔy
0 = (9800 m/s)2 + 2(-9.8 m/s2)(Δy)
Δy = 4,900,000 m

Attempt 2

1/2mv2 = mgh
h = 4,900,000 m

Neither of these equations worked, so I tried using gravitational equations, but I couldn't figure it out.

Gold Member
Do you know calculus?
What's the actual answer given in the text book?

Do you know calculus?
What's the actual answer given in the text book?

Unfortunately, I do not know calculus, and there is no answer listed in the back of the book.

Gold Member
Unfortunately, I do not know calculus, and there is no answer listed in the back of the book.

Your answer is correct if we assume that acceleration is constant.Who said it's wrong?

If you don't know calculus, then you should not bother about the varying acceleration..

Last edited:
I typed it into an online program called CAPA, and it replied that my answer was incorrect. I just emailed my professor, and he said that I need to use the full gravitational potential energy equation because the gravitational force is not constant. I've tried a few more attempts, but none of them have been correct.

Gold Member
I typed it into an online program called CAPA, and it replied that my answer was incorrect. I just emailed my professor, and he said that I need to use the full gravitational potential energy equation because the gravitational force is not constant. I've tried a few more attempts, but none of them have been correct.

CAF123
Gold Member
Yes, the value you get for Δy should tell you to use the full gravitational force equation since the value you get is of the order of magnitude of the radius of the earth. So the force will be a function of height and you cannot use the approximation U=mgh.

Sorry, just got back home.

First, I tried using this equation: 1/2mv2 = (Gm1m2/r2)Δy
Δy = (v2 * r2)/2Gm1
Δy = (98002* 63800002) / (2*(6.67*10-11)* (5.97*1024))
Δy= 4908664 m

Next, I tried omitting the Δy, instead using: r = sqrt(Gm/v2)
r = sqrt((6.67*10-11)* (5.97*1024)/ 98002)
r = 2036 m

Yes, the value you get for Δy should tell you to use the full gravitational force equation since the value you get is of the order of magnitude of the radius of the earth. So the force will be a function of height and you cannot use the approximation U=mgh.

How can I set up my equation so that force is a function of height? I've attempted to work in G and Earth's mass and radius, but I attained an answer too close to my previously found incorrect answers.

CAF123
Gold Member
Your first attempt above is incorrect because you are again using the fact that F=mg, which is only true when the height of projectile << radius of earth. Formally, to show that we do a Taylor expansion.

I am not quite sure what you did in the second attempt by simply throwing away some terms.

The method I had in mind does involve calculus as adjacent alluded to. The starting point is to write $$m_1 \ddot{z} = -\frac{Gm_1 m_2}{r^2} = -\frac{Gm_1 m_2}{(r_e + z)^2}$$ and then to do some chain rule and integration.

Your first attempt above is incorrect because you are again using the fact that F=mg, which is only true when the height of projectile << radius of earth. Formally, to show that we do a Taylor expansion.

I am not quite sure what you did in the second attempt by simply throwing away some terms.

The method I had in mind does involve calculus as adjacent alluded to. The starting point is to write $$m_1 \ddot{z} = -\frac{Gm_1 m_2}{r^2} = -\frac{Gm_1 m_2}{(r_e + z)^2}$$ and then to do some chain rule and integration.

Hmm... How would I incorporate the velocity of the object then?

I tried using 1/2mv2 = -(Gm1m2/r), but ended up getting a very small number.
98002 = - (2*(6.67*10-11)*(5.97*1024))/(6.4*106 - h)2

I ended up getting 6402880 m, but I don't want to enter it into the program because I have used almost all of my attempts already.

verty
Homework Helper
The basic idea you want to use is that kinetic energy + potential energy is conserved. Use the correct potential function and it should all fall into place. I'm not a physicist but this is surely how it works.

I think this should not be complicated.

The basic idea you want to use is that kinetic energy + potential energy is conserved. Use the correct potential function and it should all fall into place. I'm not a physicist but this is surely how it works.

I think this should not be complicated.

I know it shouldn't be complicated, and that's making it all the more frustrating. I can't seem to attain the correct answer after spending hours on this one question. I only have two attempts left.

I tried doing 1/2mv2 + mgh = - (GmM/r), so that
1/2(9800)2 + (9.8 m/s2)h = -((6.67*10-11)(5.97*1024)/6380000
h = 1.13 * 107 m (but that is incorrect as well)

I would really appreciate if someone could hold my hand and take me through this because I have no clue what to do.

Doc Al
Mentor
I tried doing 1/2mv2 + mgh = - (GmM/r), so that
You're mixing up two equations for the the potential energy. Stick to the full one. Hint: r will vary from Rearth to Rearth+h.

1 person
Orodruin
Staff Emeritus
Homework Helper
Gold Member
Try using the full expression for the potential both for the initial and for the turning point. At what r are you at the initial point? At what r are you at the turning point?

I know it shouldn't be complicated, and that's making it all the more frustrating. I can't seem to attain the correct answer after spending hours on this one question. I only have two attempts left.

I tried doing 1/2mv2 + mgh = - (GmM/r)

mgh is the potential energy above the surface if the force of gravity is assumed constant, so you can't use that.

-(gmM/r) is the potential energy of the bullet compared to the potential energy at infinity.
You need the potential energy with respect to the ground.

1 person
Oooh... So the equation needs to be set up as 1/2mv2 + (-GmM/r) = (-GmM/r+h)

Thank you to everyone who helped!!! I really appreciate your cooperation with such a dunce as myself! :p