Motion in A Plane

Two particles of masses m₁ and m₂ in projectile motion have velocities v₁ and v₂ (v₁ < v₂) respectively at t = 0. They collide at time t₀. Their velocities become  and $\overline{\mathrm{v}{\text{'}}_2}$ at time 2t₀ while still moving in it. The value of $\left|\left({\mathrm{m}}_1\overline{\mathrm{v}\text{'}} + {\mathrm{m}}_2\overline{\mathrm{v}{\text{'}}_2}\right) - \left({\mathrm{m}}_1\overline{{\mathrm{v}}_1} + {\mathrm{m}}_2\overline{{\mathrm{v}}_2}\right)\right|$ is

• zero

• (m₁ + m₂) gt₀

• 2 (m₁ + m₂) gt₀

• $\frac{1}{2}\left({\mathrm{m}}_1 + {\mathrm{m}}_2\right) {\mathrm{gt}}_0$

A block of mass m is placed on a smooth inclined plane of inclination with the horizontal. The force exerted by the plane on the block has a magnitude

• mg

• mg/cos θ

• mg cos θ

• mg tan θ

If P and Q are two non-collinear unit vectors and if $\left|\mathrm{P} + \mathrm{Q}\right| = \sqrt{3}$ then the value of (P - Q)(2P + Q) will be

• 1

• 2

• $\frac{1}{2}$

• $\frac{3}{2}$

What is the property of two vectors A and B, if$\left|\mathrm{A} + \mathrm{B}\right| = \left|\mathrm{A} - \mathrm{B}\right|$

• A and B are parallel to each other

• A and B are perpendicular to each other

• A and B are antiparallel to each other

• A and B are unit vectors

A heavy particle hanging from a fixed point by a light inextensible string of length l is projected horizontally with speed $\sqrt{\mathrm{gl}}$. When the tension in the string is equal to the weight of the particle, then the speed of particles and the inclination of the string to the vertical at the instant of motion are

• $\sqrt{\frac{\mathrm{gl}}{2}} , {\cos }^{-1}\left(\frac{1}{2}\right)$

• $\sqrt{\mathrm{gl}} , {\cos }^{-1}\left(\frac{3}{2}\right)$

• $\sqrt{\frac{\mathrm{gl}}{3}} , {\cos }^{-1} \left(\frac{2}{3}\right)$

• $\sqrt{\mathrm{gl}} , {\cos }^{-1} \left(\frac{2}{3}\right)$

A uniform circular disc of mass 1.5 kg and radius 0.5 m is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude F = 0.5 N are applied simultaneously along the three sides of an equilateral triangle xyz with its vertices on the perimeter of disc. After one second applying the forces, the angular speed of the disc in rad/s is

• 4

• 2

• 6

• 8

The escape velocity of a projectile on the earth's surface is 11.2 kms^-1 . A body is projected out with the thrice this speed. What is the speed of the body far away from the earth ? Ignore the presence of the sun and other planets. 

• 22.4 kms^-1

• 6.5 kms^-1

• 31.68 kms^-1

• 18.32 kms^-1

Two body are standing at the ends A and B of a ground, where AB = a . The boy at B starts running in a direction perpendicular to AB with velocity v₁ . The boy at A starts running simultaneously with velocity v and catches the other boy in a time t, where t is

• $\frac{\mathrm{a}}{\sqrt{{\mathrm{v}}^2 + {{\mathrm{v}}_1}^2}}$

• $\frac{{\mathrm{a}}^2}{\sqrt{{\mathrm{v}}^2 - {{\mathrm{v}}_1}^2}}$

• $\frac{\mathrm{a}}{\mathrm{v} - {\mathrm{v}}_1}$

• $\frac{\mathrm{a}}{\mathrm{v} + {\mathrm{v}}_1}$

A body of mass M at rest explodes into three pieces, two of which are of mass $\frac{\mathrm{M}}{4}$ each are thrown off in perpendicular directions with velocities of 3m/s and 4m/s respectively. The third piece will be thrown off with a velocity of

• 1.5 m/s

• 2.0 m/s

• 2.5 m/s

• 3.0 m/s

A sphere of mass m, moving with velocity v, enters in a hanging bag of sand and stops. If the mass of the bag is M and it is released by height h, then the velocity of the sphere was

• $\frac{\mathrm{M}}{\mathrm{m}}\sqrt{2\mathrm{gh}}$

• $\frac{\mathrm{m}}{\mathrm{M} + \mathrm{m}}\sqrt{2\mathrm{gh}}$

• $\frac{\mathrm{M} + \mathrm{m}}{\mathrm{m}}\sqrt{2\mathrm{gh}}$

• $\frac{\mathrm{m}}{\mathrm{M}}\sqrt{2\mathrm{gh}}$