(7) In a local bar, a customer slides an empty beer mug down the counter for a r

Question

(7) In a local bar, a customer slides an empty beer mug down the counter for a refill. The height of the counter is h. The mug slides off the counter and strikes the floor at distance d from the base of the counter.
(a) With what velocity did the mug leave the counter? (Use any variable or symbol stated above along with the following as necessary: g.)
vxi =

(b) What was the direction of the mug's velocity just before it hit the floor? (Use any variable or symbol stated above as necessary.)
? = below the horizontal

Explanation / Answer

Initial Horizontal speed = Vxi
Initial Vertical speed = 0

Vertical Distance mug travels = Height of Counter = h
Horizontal Distance mug travels = distance from the base of the counter = d

Horizontal Acceleration = 0
Vertical Acceleration =9.8 m/s^2 downwards

Time taken to carry Horizontal Distance = Time taken to carry Vertical Distance

Time taken to carry Horizontal Distance =
S = u*t + 0.5 *at^2
d = Vxi *t ----------------1

Time taken to carry Vertical Distance =
s = u *t + 0.5 * a *t^2
h = 0 + 0.5*g*t^2
h = (g/2) t^2
t = sqrt(2h/g) ------------2

Substituting t from 2 in 1 -
Vxi = d/sqrt(2h/g)

Just befor Horizontal speed u = Vxi = d/sqrt(2h/g)
Vertical speed v = g*t = g * sqrt(2h/g)

= tan^-1(g * sqrt(2h/g) / d/sqrt(2h/g)
= tan^-1( g * (2h/g)/d)
= tan^-1(2h/d)

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