Download 1-3. A ball is thrown from a tower shown in the figure. If the initial

1-3. A ball is thrown from a tower shown in the figure. If the initial velocity is 25 ft/s.
(1) Determine the x and y coordinates where the ball strikes the slope. Assume x A = 35 ft .
(2) If x A is a random variable and follows a normal distribution x A  N (35, 32 ) ft , calculate
the probability that the ball will not hit the slope.
25 ft/s
5
4
3
60 ft
1
3
35 ft
Solution
(1) Assume the ball hits the slope.
Since =
s s0 + v0t , we have
x x0 + v0 cos θ t
=
 3
0 + 25   t =
15t
=
 5
1
y=
y0 + v0 sin θ t + at 2
2
4 1
= 60 + 25   t + ( −32.2)t 2 = 60 + 20t − 16.1t 2
5 2
The equation of the slope is y − y A = k ( x − x A )
where ( x A , y A ) = (35,0) ft and k = 1 / 3 , and therefore
=
y
1
( x − 35)
3
60 + 20t − 16.1t 2=
1
(15t − 35)
3
Solving the equations above and choosing the positive root yields
t = 2.6265 s
thus
=
x 15
=
t 15(2.6265)
= 39.397 ft
y=
1
1
( x − 35) = (39.397 − 35) = 1.466 ft
3
3
Therefore, the coordinates are ( x, y ) = (39.397,1.466) ft .
Ans.
(2) When x A  N (35, 32 ) ft , the probability that the ball will not hit the slope
 x − µ xA
P( x < x A ) = 1 − P( x A < x ) = 1 − Φ 
 σx
A

 39.379 − 35 
= 1− Φ 

3


= 0.0714



Therefore, the probability that the ball will not hit the slope is 0.0714.
Ans.