Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 201412 AprilOnline
Exam
JEE Main
Year
2014
Date
12 April
Mode
Online
Subject
Physics
Chapter
Center of Mass, Momentum and Collision

JEE Main 2014 · 12 April · Online · Physics · Center of Mass, Momentum and Collision

Three masses m, 2 ~m and 3 ~m are moving in x-y plane with speed 3...

Three masses m,2 m\mathrm{m}, 2 \mathrm{~m} and 3 m3 \mathrm{~m} are moving in x−y\mathrm{x}-\mathrm{y} plane with speed 3u,2u3 \mathrm{u}, 2 \mathrm{u} and u\mathrm{u} respectively as shown in figure. The three masses collide at the same point at P\mathrm{P} and stick together. The velocity of resulting mass will be: Three masses m, 2 ~m and 3 ~m are moving in x-y plane with speed 3 u, 2 u and u respectively as shown in figure. The

Options

  1. Au12(i^+3j^)\frac{\mathrm{u}}{12}(\hat{\mathrm{i}}+\sqrt{3} \hat{\mathrm{j}})
  2. Bu12(i^−3j^)\frac{\mathrm{u}}{12}(\hat{\mathrm{i}}-\sqrt{3} \hat{\mathrm{j}})
  3. Cu12(−i^+3j^)\frac{\mathrm{u}}{12}(-\hat{\mathrm{i}}+\sqrt{3} \hat{\mathrm{j}})
  4. Du12(−i^−3j^)\frac{\mathrm{u}}{12}(-\hat{\mathrm{i}}-\sqrt{3} \hat{\mathrm{j}})

Correct answer

D. u12(−i^−3j^)\frac{\mathrm{u}}{12}(-\hat{\mathrm{i}}-\sqrt{3} \hat{\mathrm{j}})

Step-by-step solution

From the law of conservation of momentum we know that, \mathrm{m}}_{1} \mathrm{u}}_{1}+\mathrm{m}}_{2} \mathrm{u}}_{2}+\ldots .=\mathrm{m}}_{1} \mathrm{v}}_{1}+\mathrm{m}}_{2} \mathrm{v}}_{2}+\ldots . Given m1=m,m2=2 m\mathrm{m}_1=\mathrm{m}, \mathrm{m}_2=2 \mathrm{~m} and m3=3 m\mathrm{m}_3=3 \mathrm{~m} and u1=3u,u2=2u\mathrm{u}_1=3 \mathrm{u}, \mathrm{u}_2=2 \mathrm{u} and u3=u\mathrm{u}_3=\mathrm{u} Let the velocity when they stick =v→ Then, according to question, <imgsrc="/api/proxy−image?url=https=\overrightarrow{\mathrm{v}}\text{ Then, according to question, }<img src="/api/proxy-image?url=https%3A%2F%2Ffirebasestorage.googleapis.com%2Fv0%2Fb%2Fquantrexacademy-app.firebasestorage.app%2Fo%2Fquestions%252Ffigs%252Fpyq%252Fjee_main%252FjEQCngCgi4yShdpnTLkWkjmltENEJhhFn2H1vqXrJjI.original.fullsize.png%3Falt%3Dmedia&clean=1&v=wm2" data-qx-orig-src="https://firebasestorage.googleapis.com/v0/b/quantrexacademy-app.firebasestorage.app/o/questions%2Ffigs%2Fpyq%2Fjee_main%2FjEQCngCgi4yShdpnTLkWkjmltENEJhhFn2H1vqXrJjI.original.fullsize.png?alt=media" data-qx-storage-src="https://firebasestorage.googleapis.com/v0/b/quantrexacademy-app.firebasestorage.app/o/questions%2Ffigs%2Fpyq%2Fjee_main%2FjEQCngCgi4yShdpnTLkWkjmltENEJhhFn2H1vqXrJjI.original.fullsize.png?alt=media"> \\m×3u(i^)+2m×2u(−i^cos⁡60∘−j^sin⁡60∘)+3 m×u(−i^cos⁡60∘+j^sin⁡60∘)=(m+2 m+3 m)v→⇒3mui^−4mui^2−4mu(32j^)−3mui^2+3mu(32j^)=6 mv⇒mui^−32mui^−32mj^=6 mv⇒−12mui^−32muj^=6 mv⇒v→=u12(−i^−3j^)\begin{aligned}&m \times3 u(\hat{i})+2 m \times2 u\left(-\hat{i} \cos60^{{}^{\circ}}-\hat{j} \sin 60^{{}^{\circ}}\right) \\ &+3 \mathrm{~m} \times \mathrm{u}\left(-\hat{\mathrm{i}} \cos60^{{}^{\circ}}+\hat{\mathrm{j}} \sin 60^{{}^{\circ}}\right) \\ &=(\mathrm{m}+2 \mathrm{~m}+3 \mathrm{~m}) \overrightarrow{\mathrm{v}}\\ &\Rightarrow 3 m u \hat{\mathrm{i}}-4 \mathrm{mu} \frac{\hat{\mathrm{i}}}{2}-4 \mathrm{mu}\left(\frac{\sqrt{3}}{2} \hat{\mathrm{j}}\right)-3 \mathrm{mu} \frac{\hat{\mathrm{i}}}{2}\\ &+3 m u\left(\frac{\sqrt{3}}{2} \hat{\mathrm{j}}\right)=6 \mathrm{~m} \mathrm{v}\\ &\Rightarrow m u \hat{i}-\frac{3}{2}m u \hat{i}-\frac{\sqrt{3}}{2}m \hat{j}=6 \mathrm{~m} \mathrm{v}\\ &\Rightarrow-\frac{1}{2}m u \hat{i}-\frac{\sqrt{3}}{2}m u \hat{j}=6 \mathrm{~m} \mathrm{v}\\ &\Rightarrow \overrightarrow{\mathrm{v}}=\frac{\mathrm{u}}{12}(-\hat{\mathrm{i}}-\sqrt{3} \hat{\mathrm{j}})\\ & \end{aligned}
Practice Center of Mass, Momentum and Collision on Quantrex Academy →

More from Center of Mass, Momentum and Collision

Two identical bodies A and B of equal masses have initial velocities v_1 = 4 i m/s and… JEE Main 2026The position of center of mass of three masses 2 kg, 3 kg and 15 kg placed with respect… JEE Main 2026Two blocks of masses 2 kg and 1 kg respectively, are tied to the ends of a string which… JEE Main 2026The position of an object having mass 0.1 kg as a function of time t is given as r =… JEE Main 2026A body of mass 14 kg initially at rest explodes and breaks into three fragments of masses… JEE Main 2026In a perfectly inelastic collision, two spheres made of the same material with masses 15… JEE Main 2026Given below are two statements : Statement I : For a mechanical system of many particles… JEE Main 2026In a scattering experiment, a particle of mass 2 m collides with another particle of mass… JEE Advanced 2025 Full Center of Mass, Momentum and Collision list All JEE Main PYQs