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Projectile motion proof 
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#1
Jul2906, 09:21 AM

P: 738

If m1 is thrown with an angle of [tex]\theta[/tex] from the ground at the same time m2 drops in free motion from the top of a ceiling, prove that
(1) If m1 aims at m2, as long as the initial velocity is large enough, m1 will hit m2 (2) The inital velocity V must be larger than [tex][gR/(sin2\theta]^1/2[/tex] My calculations: for m1: 1. R=Vcos[tex]\theta[/tex]t 2. y1=Vsin[tex]\theta[/tex]t0.5g[tex]t^2[/tex] for m2: y2=0.5g[tex]t^2[/tex] but i'm stuck on what to do next... 


#2
Jul2906, 10:15 AM

Mentor
P: 41,323

First fix your equation for y2. Like m1, m2 is falling. Hint: Assume the initial position of m2 is x_0, y_0 (where m1 starts at 0,0).



#3
Jul2906, 10:16 AM

P: 262

After you fix what Doc Al said, you have started correctly. Now think about what the question is asking you.
When m1 hits m2, what do you know about both of their final positions? What does this tell you about the final (x, y) position? To keep things straight, I would use x instead of R in your first equation, but that's up to you. If you get the first part, the second part shouldn't be too difficult. Just remember the trigonometric identity that [tex]2\sin{\theta}\cos{\theta}=\sin{2\theta}[/tex]. Once you get the physics part of it correctly, then the rest of it is math. 


#4
Jul2906, 10:22 AM

P: 6

Projectile motion proof
is r the range of the projectile??
however your first question is easily answered. both the bodies will have the same y displacement . so the projectile body will always hit the m2 body 


#5
Jul2906, 09:39 PM

P: 262




#6
Jul3006, 02:27 AM

P: 738

So my mistake is that for m1 and m2 they have different coordinate references, so that's why i can't use the equations for m1 and m2 together... is that right?



#7
Jul3006, 04:20 AM

Math
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PF Gold
P: 39,361




#8
Jul3006, 11:29 AM

P: 262

Think about the physical significance of the equations that you have derived. [tex]x = v\cos{\theta}t[/tex]  What does this mean about the fired projectile's motion? Notice that I have changed the R to an x, so that we can keep the coordinates straight. [tex]y_1 = v_0\sin{\theta}t\frac{1}{2}gt^2[/tex]  What does this mean about the fired projectile? Is it falling or rising, and which way is it accelerating? [tex]y_2 = \frac{1}{2}gt^2[/tex]  Compare this equation to your [tex]y_1[/tex] equation. According to what you have derived here, which way is the ball accelerating? Check to see that your equations make sense to you before you start solving for the unknowns. In this case, both objects are falling, so their accelerations are downward. 


#9
Jul3106, 04:06 AM

P: 738

ok!
[tex]y_2 = \frac{1}{2}gt^2[/tex] so y1=y2 => Vsin[tex]\theta[/tex]*t0.5g[tex]t^2[/tex]=[tex]\frac{1}{2}gt^2[/tex] =>Vsin[tex]\theta[/tex]*t=0 ? 


#10
Jul3106, 05:51 AM

P: 167

(Referring to Doc Al's post) For this question, we set the initial position of [tex]m_{1}[/tex] as the origin. In addition, we assume [tex]m_{2}[/tex] starts from the point [tex](x_{0}, y_{0})[/tex].
When considering the vertical displacement, we will need a reference. In this case, we set it to be the xaxis. Hence, all measurements of vertical displacement will be with respect to this line. Since [tex]m_{2}[/tex] already has some vertical displacement to begin with, the equation you need should be [tex]y_2 = y_{0}\frac{1}{2}gt^2[/tex] P/S In the second part of the question, you need to show that the initial velocity V must be bigger than [tex](\frac{gR}{sin2\theta})^{1/2}[/tex]. What is R? 


#11
Aug106, 04:48 AM

P: 738

ok! thank you!
R is the horizontal length from the point m1 is at rest to where m2 falls on the ground... 


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