Answer:
a) the vector equation is r(x,y) = (35cos27°t, 16t^2 -35sin27° - 4)
b) Maximum height = 7.945ft
c) Time of flight = 1.201 secs and range = 37.45ft
d) when t = 0.74, the distance above the ground = 14.37ft
When t = 0.254, the distance above the ground = 26.53ft
e) if the ball is raised to 8ft, the ball won't be able to reach it. This is because the maximum height is 7.954ft
Step-by-step explanation:
a) From the diagram
x0 = 0
y0= 4
V0(initial velocity) = 35
α= 27°
y = 4 + 35sin27°t - 16t^2
y = -16t^2 + 35sin27°t+ 4
y = 16t^2 - 35sin27°t - 4
x = 35cos27°t
Vector equation for the path of the volleyball = r(x,y)
r(x,y) = (35cos27°t, 16t^2 -35sin27° - 4)
b) the ball reaches maximum when dy/dt = 0
y = 16t^2 - 35sin27°t - 4
dy/dt = 32 - 35sin27°
0 = 32 - 35sin27°t
-32 = -35sin27°t
t = -32 / -35sin27°
t = 0.4966 seconds
Put t = 0.4966 into y = 16t^2 - 35sin27°t - 4
y = 16(0.4966)^2 - 35sin27°(0.4966) - 4
y = 7.945 ft
The maximum height is 7.945 ft
c) To find the range and flight time, put y= 0
0 = 16t^2 - 35sin27°t - 4
0 = 16t^2 - 15.89t - 4
Using quadratic equation general formula,
[-b +/- √b^2 -4ac] / 2a
a = 16, b = -15.89, c = -4
= [-(-15.89) +/- √ (-15.89)^2 - 4(16)(-4)] /2(16)
= [15.89 +/- √(15.89)^2 + 256] / 32
= (15.59 +/- 22.55) / 32
= (15.89 + 22.55) / 32 or (15.89 - 22.55) / 32
= 1.201 or -0.208
Time of flight = 1.201 secs
Range = x = 35cos27°t
Range = 32cos27°(1.201)
= 37.45 ft
d) when the volley is 7ft above ground, y = 7
Recall that y = 16t^2 - 35sin27°t - 4
7 = 16t^2 - 35sin27°t - 4
0 = 16t^2 - 35sin27°t - 4 +7
0 = 16t^2 - 35sin27°t + 3
0 = 16t^2 - 15.89t + 3
Using quadratic equation general formula,
[-b +/- √b^2 -4ac] / 2a
a = 16, b = -15.89, c = 3
= [-(-15.89) +/- √ (-15.89)^2 - 4(16)(3)] /2(16)
= [15.89 +/- √(15.89)^2 - 192] / 32
= (15.59 +/- 7.7778) / 32
= (15.89 + 7.7778) / 32 or (15.89 - 7.7778) / 32
= 0.74 or 0.254
When t = 0.74,
x = 35cos27°t
x = 35cos27°(0.74)
x = 23.08 ft
Therefore , R - x(0.74)
= 37.45 - 23.08
= 14.37ft
When t = 0.254,
x = 35cos27°t
x = 35cos27°(0.254)
x = 7.92 ft
Therefore R - x(0.254) =
37.45 - 7.92 = 29.53ft
e) if the ball is raised to 8ft, the ball won't be able to reach it. This is because the maximum height is 7.954ft
