Lat eli) e kl, calculate the unit tanannt vector. Three-dimensional motion Consi

Question

Lat eli) e kl, calculate the unit tanannt vector. Three-dimensional motion Consider the motion of the follo ing objects. Assume the x-axis points east, the y-axis points north, t positive z-axis is vertical and opposite g. the ground is horizontal, a only the gravitational force acts on the object unless otherwise stat a. Find the velocity and position vectors for, t 2 0. b Determine the time offtight and range of the object. Determine the maximum height of the object. A baseball is hit 3 ft above home plate with an initial velocity of K30. 30, 80) ft/s. The spin on the baseball produces a borizontal acceleration of the ball of 5 ft/s in the northward direction.

Explanation / Answer

22)

We use the kinematic equations of motions in each of the three directions,i.e

v=u+at, s=ut+(1/2)at2 , and v2=u2+2as

Given ax=0, ay=5, az=-32.174

vx=ux+axt

vx=30

vy=30+5t and vz=80-32.174t

Similarly for position vectors

Sx=uxt+(1/2)axt2=30t ,

Sy= 30t+(1/2)(5)t2 , and Sz= 80t-(1/2)(32.174)t2

Now, the flight ends when the ball reaches the ground, or in other words Sz=-3

Substituting Sz=-3 we get,

-3=80t-(1/2)(32.174)t2

Therefore t=5.01,-0.037

since time cant be negative time of flight t=5.01s.

Substituting this t in Sx and Sy we get Sx=150.3ft and Sy=213.05ft which is the required range.

Now,

maximum height will be reached when vz=0

Using  v2=u2+2as

we get, 0=uz2+2azS

Sz=-uz2/2az = -(802)/(2)(-32.174) = 99.456ft

Therefore maximum height will be 99.456+3=102.459ft. as it started from 3ft height.

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