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Motion in A Plane question

2021 · 31 Aug · Shift 1 · Q46
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  5. /2021 · 31 Aug · Shift 1 · Q46

Motion in A Plane question

2021 · 31 Aug · Shift 1 · Q46

JEE MainPhysicsMotion in A PlaneMCQ+4 / −1
A helicopter is flying horizontally with a speed 'v' at an altitude 'h' has to drop a food packet for a man on the ground. What is the distance of helicopter from the man when the food packet is dropped?
  1. A
    2ghv2+1h2\sqrt {{{2gh{v^2} + 1} \over {{h^2}}}}h22ghv2+1​​
  2. B
    2ghv2+h2\sqrt {2gh{v^2} + {h^2}}2ghv2+h2​
  3. C
    2v2hg+h2\sqrt {{{2{v^2}h} \over g} + {h^2}}g2v2h​+h2​
  4. D
    2ghv2+h2\sqrt {{{2gh} \over {{v^2}}} + {h^2}}v22gh​+h2​
View written solutionFree

Correct answer: C

  1. Understand the motion of the packet

When the food packet is dropped from the helicopter:

  • It has initial horizontal velocity equal to the helicopter's speed, i.e. vvv.
  • Its initial vertical velocity is 000.
  • It falls freely under gravity from height hhh.

So this is a projectile motion problem.

  1. Time taken to fall from height hhh

For vertical motion,

h=12gt2 h = \frac{1}{2}gt^2h=21​gt2

So,

t=2hg t = \sqrt{\frac{2h}{g}}t=g2h​​
  1. Horizontal distance traveled in this time

Since horizontal velocity remains constant at vvv,

x=vt=v2hg x = vt = v\sqrt{\frac{2h}{g}}x=vt=vg2h​​

This is the horizontal distance between the man and the point vertically below the helicopter at the instant of release.

  1. Actual distance of helicopter from the man at the moment of dropping

At the instant of dropping:

  • Vertical separation between helicopter and man = hhh
  • Horizontal separation = x=v2hgx = v\sqrt{\frac{2h}{g}}x=vg2h​​

Hence, by Pythagoras,

d=x2+h2 d = \sqrt{x^2 + h^2}d=x2+h2​

Substitute x2=v2⋅2hgx^2 = v^2 \cdot \frac{2h}{g}x2=v2⋅g2h​:

d=2v2hg+h2 d = \sqrt{\frac{2v^2h}{g} + h^2}d=g2v2h​+h2​
  1. Match with options

This matches:

2v2hg+h2\boxed{\sqrt{\frac{2v^2h}{g} + h^2}}g2v2h​+h2​​

which is Option C.

  1. Comparison with stored correct answer

Stored correct answer: C

Our derived answer: C

So they agree.

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