- #1

- 58

- 0

__Physics for Scientists and Engineers__by Paul A. Tipler, 4th edition. The statement of the problem is as follows:

**Ball A is dropped from the top of a building at the same instant that ball B is thrown vertically upward from the ground. When the balls collide, they are moving in opposite directions, and the speed of A is twice the speed of B. At what fraction of the height of the building does the collision occur?**

My incomplete [<Ahem> Make that: COMPLETE] attempt at a solution goes as follows:

We denote the height of the building as

**h**, the position of ball A as

**y**, the initial position of ball A as

_{A}**y**, and analogously for ball B.

_{A0}The equation of motion for ball A is the following:

**y**

y

v

g = 9.81 m/s

Therefore,

_{A}- y_{A0}= v_{A0}t - (0.5)gt^{2}y

_{A0}= hv

_{A0}= 0g = 9.81 m/s

^{2}**y**

=> y

_{A}= -(0.5)(9.81 m/s^{2})t^{2}+ h=> y

_{A}= -4.905t^{2}+ hThe equation of motion for ball B is the following:

**y**

y

g = 9.81 m/s

Therefore,

_{B}- y_{B0}= v_{B0}t - (0.5)gt^{2}y

_{B0}= 0g = 9.81 m/s

^{2}**y**

=> y

_{B}= v_{B0}t - (0.5)(9.81 m/s^{2})t^{2}=> y

_{B}= v_{B0}t - 4.905t^{2}The two balls collide when

**y**:

_{A}= y_{B}**y**

=> h = v

_{A}= y_{B}=> -4.905t^{2}+ h = v_{B0}t - 4.905t^{2}=> h = v

_{B0}tFor ball A:

**v**

v

Therefore,

_{A}= v_{A0}- gtv

_{A0}= 0**v**

_{A}= - gt => v_{A}= -9.81tFor ball B:

**v**

Therefore,

_{B}= v_{B0}- gt**v**

_{B}= v_{B0}- gt => v_{B}= v_{B0}- 9.81tWe also know that when

**y**,

_{A}= y_{B}**v**:

_{A}= -2v_{B}**v**

=> -9.81t = -2v

=> 2v

=> v

_{A}= -2v_{B}=> -9.81t = -2[v_{B0}- 9.81t]=> -9.81t = -2v

_{B0}+ 19.62t=> 2v

_{B0}= 29.43t=> v

_{B0}= 14.715tSince

**h = v**, we have:

_{B0}t**h = (14.715t)t = 14.715t**

^{2}Unfortunately, I couldn't figure out what to do from this point until I spent the better part of the last hour typing this thread. :grumpy: Then, as I was typing the previous sentence, it hit me:

**x**

=> x

_{A}= -4.905t^{2}+ h = -4.905t^{2}+ 14.715t^{2}=> x

_{A}= 9.81t^{2}Therefore,

**x**

=> x

=> x, which is the answer given in the back of the book.

_{A}/ h = 9.81t^{2}/ (14.715t^{2})=> x

_{A}/ h = 2/3=> x

_{A}= 2h/3I've decided to post this thread anyway, partly because someone else might benefit from seeing it, and partly because I spent nearly an hour typing it.

This also reminds me of a post by Clausius2 in a previous thread I had posted:

I don't think this would have worked the last time, but it certainly worked here! :rofl:Clausius2 said:Maybe the time you have spent writing such [an elaborate] thread you could have re-written your solution of the problem, and surely you would find the error.

Oh brother ... Good night, everyone! :zzz: