a missile is fired for maximum range at your town from a place in the enemy country 100 km away from your town . if the missile is first detected at its half way point , how much warning time will you have (take g= 10 m /sec2)
a) 100 seconds
b)100/√2s
c) 100√3/2s
d) 200s
Answers
Answer:
Range=
g
u
2
sin2θ
=
10
20∗20
=40m
because θ=90
∘
for maximum range
The correct option is d) 200s (approx).
To find,
Warning time.
Given,
100 km away.
Solution,
Let's assume that the missile is fired at an angle such that it reaches its maximum range at the halfway point. In that case, the range of the missile would be:
R = (v^2/g) * sin(2θ)
where v is the velocity of the missile, g is the acceleration due to gravity, and θ is the angle of elevation of the missile.
Since the missile is fired for maximum range, we can set dR/dθ = 0 to find the value of θ that maximizes R:
dR/dθ = (2v^2/g) * cos(2θ) = 0
cos(2θ) = 0
2θ = π/2
θ = π/4
Substituting this value of θ in the equation for R, we get:
R = (v^2/g) * sin(π/2) = (v^2/g)
Since the missile is detected at the halfway point, the range of the missile would be:
R/2 = (v^2/g)/2
We know that R/2 = 100 km, so we can solve for v:
100000 = (v^2/g)/2
v^2/g = 200000
v = √(200000 * 10) = 200√2 m/s
The warning time can be calculated as the time taken by the missile to cover the remaining 50 km (half of its maximum range) at its velocity:
t = d/v = 50000/(200√2) = 100/√2 s
Therefore, the answer is (b) 100/√2 s.
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