A small ball is projected from the…

A small ball is projected from the foot of a smooth fixed incline of inclination 30^° with speed 20 m s^-1, in the vertical plane containing the line…

One or more options correct

A small ball is projected from the foot of a smooth fixed incline of inclination

30^\circ

with speed

20\ \mathrm{m\,s^{-1}}

, in the vertical plane containing the line of greatest slope. The velocity of projection makes an angle of

30^\circ

with the incline, above it and directed up the slope. Take

g=10\ \mathrm{m\,s^{-2}}

and neglect air resistance. Choose the correct statement(s).

A. The time of flight, until the ball again meets the incline, is $\dfrac{4}{\sqrt3}\ \mathrm{s}$ .
B. The distance from the point of projection to the point where the ball strikes the incline is $\dfrac{80}{3}\ \mathrm{m}$ .
C. The ball strikes the incline with its velocity perpendicular to the incline.
D. The maximum perpendicular distance of the ball from the incline is $5\ \mathrm{m}$ .

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Answer: A, B

Use axes along and perpendicular to the incline. Here

g_\parallel=g\sin30^\circ=5

(down the slope) and

g_\perp=g\cos30^\circ=5\sqrt3

. The initial components are

u_\parallel=20\cos30^\circ=10\sqrt3

and

u_\perp=20\sin30^\circ=10

. Time of flight:

T=\dfrac{2u_\perp}{g_\perp}=\dfrac{20}{5\sqrt3}=\dfrac{4}{\sqrt3}\ \mathrm{s}

, so (A) is true. Range along the slope:

R=u_\parallel T-\tfrac12 g_\parallel T^2=10\sqrt3\cdot\dfrac{4}{\sqrt3}-\tfrac52\cdot\dfrac{16}{3}=40-\dfrac{40}{3}=\dfrac{80}{3}\ \mathrm{m}

, so (B) is true. At impact the component along the slope is

u_\parallel-g_\parallel T=10\sqrt3-\dfrac{20}{\sqrt3}=\dfrac{10}{\sqrt3}\neq0

, so the velocity is not perpendicular to the incline: (C) is false. Maximum perpendicular distance

H=\dfrac{u_\perp^2}{2g_\perp}=\dfrac{100}{10\sqrt3}=\dfrac{10}{\sqrt3}\approx5.77\ \mathrm{m}

, not

5\ \mathrm{m}

(that would come from using

g

instead of

g\cos30^\circ

): (D) is false.

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Original question written in the JEE Advanced 2020 Paper 2 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake