a. Find an expression for the initial launch speed v0 when the ball emerges hori

Question

a. Find an expression for the initial launch speed v0 when the ball emerges horizontally from the machine. Your answer could depend on h, D, m, R and any relevant physical constants.

b. Now find an expression for the magnitude of the normal force FN exerted by the curved part of the machine on the ball just before the ball is launched horizontally from the tube. Your answer could depend on h, D, m, R, and any relevant physical constants. (Note: If you could not find an expression for the speed v0 in part (a), you may express your answer here in terms of v0.)

c. Find an expression for the minimum launch speed vmin such that the ball will emerge horizontally from the pitching machine. Your answer could depend on m, R, and any relevant physical constants. (Hint: If the ball doesn’t emerge horizontally it won’t be in contact with the machine at the top of the curved metal arc.)

Explanation / Answer

at thepoint of emergence

along vertical

initial velocity = voy = 0

displacement y = -h

from equation of motion

y = voy*T + 0.5*ay*T^2

-h = -0.5*g*T^2

T =sqrt(h/g)

along horizantal

D = vo*T

vo = D/T = D*sqrt(g/2h) <<<<-------------answer

++++++++++++++

(b)

Fn = m*vo^2/R = m*D^2*(g/2hR) = m*D^2*g/2hR

++++++++++++++++++++++

(c)

kinetic energy at launch K1 = 0.5*m*vmin^2

at the emerge point

K2 = 0.5*m*vo^2

U2 = m*g*h

from energy conservation

K1 = K2 + U2

0.5*m*vmin^2 = 0.5*m*vo^2 + m*g*h

0.5*vmin^2 = 0.5*D^2*g/2h + gh

vmin = sqrt((D^2g/2h)+2gh)

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