A heavy particle is projected with a velocity at an angle with the horizontal into a uniform gravitational field. The slope of the trajectory of the particle varies not according to which of the following curve?
A
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B
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C
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D
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Solution
The correct options are B C D Equation of the trajectory for the projectile, y=xtanθ−gx22u2cos2θ Slope of the trajectory, dydx=tanθ−2gx2u2cos2θ The above equation can be compared with the straight line equation, y=mx+c where c=tanθ and m=−2g2u2cos2θ Since both options B and D have positive slope for variation with x. Hence the slope of trajectory does not vary according to the variation shown by option B and D Vertical component of velocity using 1st equation of motion, Vy=usinθ−gt(2) Horizontal component of velocity, Vx=ucosθ(3) Dividing (2) and (3) Slope=VyVx=tanθ−gtucosθ The above equation matches with the straight line equation between slope and time(t) i.e y=−mx+c This implies option C also does not show the variation of slope of trajectory since the curve shows either quadratic or cubic variation. Hence correct options are B,C and D