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matematikawan
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I thought I have already master the topic on projectile. Now I'm not that sure. I fail to solve the Problem 3.4 in the book An Introduction to Mechanics by Daniel Kleppner and Robert J. Kolenkow.
The verbatim problem is as follows:
"An instrument-carrying projectile accidentally explodes at the top of its trajectory. The horizontal distance between the launch point and the point of explosion is L. The projectile breaks into two pieces which fly apart horizontally. The large piece has three times the mass of the smaller piece. To the surprise of the scientist in charge, the smaller piece returns to Earth at the launching station.
How far away does the larger piece land? Neglect air resistance and effects due to the earth’s curvature."
I have consider something like this:
Assume that the velocity of the projectile before the explosion to be [tex]u\hat{i}[/tex] (top of the trajectory). Let the velocities of the pieces after the explosion be
[tex]\vec{V}_1 = V_{1x}\hat{i} + V_{1y}\hat{j} [/tex] and [tex]\vec{V}_2 = V_{2x}\hat{i} + V_{2y}\hat{j} [/tex]. Also let the time for the smaller piece to return to the launching station be t1. Then
-L = V1xt1 ------ (1)
Now I have already introduced six variables u, V1x , V2x , V1y , V2y and t1.
I also have to know when the larger piece landed (t2 - another variable?) in order for me to calculate where the larger piece landed. That's the seventh variable.
From the principle of conservation of momentum, I can get another two equations. Altogether now I have only 3 equations with 7 unknowns.
My questions:
1. What other 4 equations can I have without introducing another unknowns.
2. Is t1 = t2 ?
3. Why is the momentum conserved in the vertical direction ? There is an external force, gravity acting downward.
Really appreciate any suggestion. Thank you for your interest.
The verbatim problem is as follows:
"An instrument-carrying projectile accidentally explodes at the top of its trajectory. The horizontal distance between the launch point and the point of explosion is L. The projectile breaks into two pieces which fly apart horizontally. The large piece has three times the mass of the smaller piece. To the surprise of the scientist in charge, the smaller piece returns to Earth at the launching station.
How far away does the larger piece land? Neglect air resistance and effects due to the earth’s curvature."
I have consider something like this:
Assume that the velocity of the projectile before the explosion to be [tex]u\hat{i}[/tex] (top of the trajectory). Let the velocities of the pieces after the explosion be
[tex]\vec{V}_1 = V_{1x}\hat{i} + V_{1y}\hat{j} [/tex] and [tex]\vec{V}_2 = V_{2x}\hat{i} + V_{2y}\hat{j} [/tex]. Also let the time for the smaller piece to return to the launching station be t1. Then
-L = V1xt1 ------ (1)
Now I have already introduced six variables u, V1x , V2x , V1y , V2y and t1.
I also have to know when the larger piece landed (t2 - another variable?) in order for me to calculate where the larger piece landed. That's the seventh variable.
From the principle of conservation of momentum, I can get another two equations. Altogether now I have only 3 equations with 7 unknowns.
My questions:
1. What other 4 equations can I have without introducing another unknowns.
2. Is t1 = t2 ?
3. Why is the momentum conserved in the vertical direction ? There is an external force, gravity acting downward.
Really appreciate any suggestion. Thank you for your interest.