A rifleman on a train moving with a speed of $$32$$ $$kmh-1$$ fires at an object running away from the train at right angle with a speed of $$24$$ kmh. The line connecting the man and the object makes an angle of $$30$$° to the train at the instant of shooting. The angle of aiming with the direction of train is α in order to hit the object. The muzzle velocity is $$450$$ $$kmh-1$$. If $$sin$$α−π$$6=n+63225$$, then find the value of n.

Question

A rifleman on a train moving with a speed of $$32$$ $$kmh-1$$ fires at an object running away from the train at right angle with a speed of $$24$$ kmh. The line connecting the man and the object makes an angle of $$30$$° to the train at the instant of shooting. The angle of aiming with the direction of train is α in order to hit the object. The muzzle velocity is $$450$$ $$kmh-1$$. If $$sin$$α−π$$6=n+63225$$,  then find the value of n.

Infinity Learn · Accepted Answer

Let A be the rifleman and P the object at the instant of firing the shot and Q, the object at the moment when it is hit by the shot. Let QP meet the path of train at M.Let the shot takes t hour t reach the object.∴Velocity of shot in the direction of $$AM=450cos$$α$$+32$$∴ $$d=450cos$$α$$+32$$           …(i)The$$ velocity of shot in direction of $$MQ=450sin$$αNow, $$PQ=24t$$∴ $$MQ=MP+PQ=dtan$$⁡$$30$$∘$$+24t$$∴ $$(450sin$$α$$)t=dtan$$ $$30$$∘$$+24t$$∴ dtan⁡$$30$$∘$$=450tsin$$ α−$$24t$$               …$$(ii)Dividing$$ Eqs. $$(i) by (ii), we $$get450cos$$ α$$+32450sin$$ α−$$24=1tan$$⁡$$30$$∘$$=3$$⇒ $$sin$$ α−$$30$$∘$$=8+63225$$∴ $$n=8$$

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