A plane is inclined at an angle \alpha = 30° with respect to the horizontal. A particle is projected with a speed u = 2 ms–1 , from the base of the plane, making an angle \theta = 15° with respect to the plane as shown in the figure. the distance from the base, at which the particle hits the plane is close to : (Take g = 10 ms –2)

Motion in A Plane PYQ — JEE Main 2019 · 10 Apr · Shift 2 · Q45 with Solution

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Correct answer: C

The plane is inclined at $\alpha = 30^\circ$ and the particle is projected at $\theta = 15^\circ$ with respect to the plane.

So, the angle of projection with respect to the horizontal is $\phi = \alpha + \theta = 30^\circ + 15^\circ = 45^\circ.$

Initial speed: $u = 2\,\text{m s}^{-1}$

Hence, projectile equation is $x = u\cos\phi\, t,$ $y = u\sin\phi\, t - \frac{1}{2}gt^2.$

With $\phi=45^\circ$ , $x = 2\cdot \frac{1}{\sqrt 2} t = \sqrt 2\, t,$ $y = \sqrt 2\, t - 5t^2.$

The plane passes through the origin and makes angle $30^\circ$ with horizontal, so $y = x\tan 30^\circ = \frac{x}{\sqrt 3}.$

The particle hits the plane again when its coordinates satisfy this equation.

Substitute $x=\sqrt2 t$ into plane equation: $y = \frac{\sqrt2 t}{\sqrt3}.$

Now equate with projectile's $y$ : $\sqrt2 t - 5t^2 = \frac{\sqrt2 t}{\sqrt3}.$

Rearrange: $\sqrt2 t\left(1-\frac{1}{\sqrt3}\right) = 5t^2.$

Ignoring the trivial solution $t=0$ , $t = \frac{\sqrt2}{5}\left(1-\frac{1}{\sqrt3}\right).$

Numerically, $\sqrt2 \approx 1.414, \qquad \frac{1}{\sqrt3} \approx 0.577.$

So,

=0.2828\times 0.423 \approx 0.1196\,\text{s}.$$ --- 4. **Find horizontal distance $x$** $$x = \sqrt2 t \approx 1.414\times 0.1196 \approx 0.169\,\text{m}.$$ --- 5. **Convert to distance along the plane** If $s$ is the distance along the incline, then $$x = s\cos 30^\circ$$ so $$s = \frac{x}{\cos30^\circ} = \frac{0.169}{\sqrt3/2}.$$ Since $$\cos30^\circ \approx 0.866,$$ we get $$s \approx \frac{0.169}{0.866} \approx 0.195\,\text{m}.$$ Thus, $$s \approx 19.5\,\text{cm}.$$ This is closest to **20 cm**. --- 6. **Option check** - A: $14$ cm ❌ - B: $18$ cm ❌ - C: $20$ cm ✅ - D: $26$ cm ❌ So the correct option is **C**.