A cart is moved horizontally with a constant velocity of 4 m/sec. A ball is thrown from it with a velocity of 4 m/sec and at an angle θ with the horizontal with respect to the cart. Assume the height of the cart is very small, so that the motion of the ball is assumed to be a ground-to-ground projectile. Horizontal range of the ball with respect to the ground is R1 and that with respect to the cart is R2. Then
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a
R1 will be maximum for θ=60∘
b
R1 will be maximum for θ=90∘
c
R2 will be maximum for θ=60∘
d
R2 will be maximum for θ=45∘
answer is A.
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Detailed Solution
Range with respect to ground R=2uxuyg=2(4+4cosθ)(4sinθ)gR=32sinθ(1+cosθ)g=32g(sinθ+sinθcosθ)Rmax when sinθ+sinθcosθ will be max ⇒ ddθ(sinθ+sinθcosθ)=0⇒ cosθ+2cos2θ−1=0⇒ cosθ=12⇒θ=60∘Since the lvelocityl of the ball w.r.t. to the cart is constant. So range w.r.t. to the cart will be max when θ = 45o