Does Bouncing a Baseball Help It Travel Faster?

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The discussion revolves around a projectile motion problem involving a baseball thrown from the outfield to home plate, specifically examining whether bouncing the ball allows it to travel faster. The initial scenario involves throwing the ball at 50 m/s at a 45-degree angle, resulting in a total distance of 255.30 meters and a time of flight of 7.22 seconds. The challenge arises when attempting to calculate the distance and time of flight for a bounce scenario, where the ball rebounds at the same angle but at half the speed (25 m/s). The user struggles to develop equations for the time of flight and angle theta for the bounce, leading to confusion over the calculations for each segment of the throw. Assistance is sought to clarify the physics involved in setting up the problem correctly.
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I'm working on projectile motion problem and it is a two-fold problem about throwing a baseball from outfield to home plate. The theory is that the ball arrives faster if thrown on a bounce. "Suppose that after the bounce the ball rebounds at the same angle theta as it had when initially released but it loses half its speed. (a) Assuming the ball is always thrown with the same initail speed, at what angle theta should the ball be thrown in order to go the same distance D with one bounce as one thrown upward at 45.0 deg. with no bounce?

I have solved the no bounce time and distance using the arbitrarily chosen value of 50 m/s as an initial velocity with the notion that I would use this same initial velocity to solve for the "bounce" problem. Using this velocity I have found that the total TOF = 7.22s and the ball travels a distance "D" of 255.30 m. So far so good.

Setting up for the "bounce" problem I am dividing the total distance "D" into two sections D-1 with an initial velocity of 50 m/s and D-2 with an initial velocity of 25 m/s. I know that I have to add D-1 and D-2 to = "D" or 255.30 meters. What I don't know is how to develop an equation to solve for either time or for the angle theta without having any further data to work with.

I'm coming up with some really odd numbers for D-1 and D-2 TOF. For instance for D-1 I am getting 10.204(sin theta) = TOF and 5.102(sin theta) for D-2 and that doesn't seem to get me where I need to be if I don't know or can't solve for theta.

Any suggestions?
 
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Well D-1 = 50*t right? Isn't D-2 just 25*(7.22-t) if t is the time of bounce? I mean I may be oversimplifying the problem but the time it takes the ball to decelerate to 25m/s probably is pretty short considering the bounce is short. The x component of the velocity should be pretty consistent along those distance intervals.
 
Please Help!

I am still having difficulty with a physics problem. The course is trig based. I can't seem to set up the problem with the baseball taking the hop.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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