A ship is Approaching a cliff of height 105m above sea level . A gun fitted on the ship can fire shots with a speed of 110m/s .Find the maximum distance from the foot of the cliff from where the gun can hit an object on the top of the cliff. [ Take g= 10m/ s^2]
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Answers
1100 m is the maximum distance from the foot of the cliff from where the gun can hit an object on the top of the cliff
Explanation:
Let say angle = α and time = t to reach peak
S = ut + (1/2)at²
=> 105 = 110Sinα * t - 5t²
=> 110Sinα * t = 105 + 5t²
Squaring both sides
=> (110Sinα)²t² = 105² + 25t⁴ + 1050t²
=> (110² - (110Cosα)²)t² = 105² + 25t⁴ + 1050t²
=> (110Cosα)² t² = 110²t² - 105² - 25t⁴ - 1050t²
=> (110Cosα)² t² = - 25t⁴ + 11050t² - 105²
Horizontal Distance
D = 110Cosα * t
Z = D² = (110Cosα)² t²
=> Z = - 25t⁴ + 11050t² - 105²
as D is + ve hence D will be maximum if D² = Z is maximum
Z = - 25t⁴ + 11050t² - 105²
Let substitute t² = x
=> Z =- 25x² + 11050x - 105²
dZ/dx = -50x + 11050
=> x = 11050/50
=> x = 221
d²Z/dx² = - 50 Hence maximum value
t² = 221
=> t = √221
D² = Z = - 25t⁴ + 11050t² - 105²
= -25(221)² + 11050(221) - 105²
= -12,21,025 + 24,42,050 - 11025
= 12,10,000
=> D = 1100 m
maximum distance from the foot of the cliff from where the gun can hit an object on the top of the cliff. = 1100 m
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