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Question

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=xtanθgx22u2cos2θ
Slope of the trajectory,
dydx=tanθ2gx2u2cos2θ
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=usinθgt(2)
Horizontal component of velocity, Vx=ucosθ(3)
Dividing (2) and (3)
Slope=VyVx=tanθgtucosθ
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


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