A particle is projected with a velocity v so that its horizontal range twice the greatest height attained. The horizontal range is

• $\frac{4{\mathrm{v}}^2}{5 \mathrm{g}}$

• $\frac{{\mathrm{v}}^2}{\mathrm{g}}$

• $\frac{{\mathrm{v}}^2}{2\mathrm{g}}$

• $\frac{2{\mathrm{v}}^2}{3 \mathrm{g}}$

Three projectiles A, B and C are projected at an angle of 30°, 45°, 60° respectively. If R_A, R_B and R_C are ranges of A, B and C respectively, then (velocity of projection is same for A, B and C)

• R_A = R_B = R_C

• R_A = R_C > R_B

• R_A < R_B < R_C

• R_A = R_C < R_B

The component ofa vector r along X-axis will have a maximum value, if

• r is along positive X-axis

• r is along positive Y-axis

• r is along negative Y-axis

• r makes an angle of 45° with the X-axis

Maximum acceleration of the train in which a 50 kg box lying on its floor will remain stationary (Given, coefficient of static friction between the box and the train's floor is 0.3 and g = 10 ms^-2)

• 5.0 ms^-2

• 3.0 ms^-2

• 1.5 ms^-2

• 15 ms^-2

Three bodies a ring (R), a solid cylinder (C) and a solid sphere (S) having same mass and same radius roll down the inclined plane without slipping. They start from rest, if v_R, v_C and v_S are velocities of respective bodies on reaching the bottom of the plane, then

• v_R = v_C = v_S

• v_R > v_C > v_S

• v_R < v_C < v_S

• v_R = v_C > v_S

If $\mathrm{A} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 8\hat{\mathrm{k}}$ is perpendicular to $\mathrm{B} = 4\hat{\mathrm{j}} - 4\hat{\mathrm{i}} + \mathrm{\alpha }\hat{\mathrm{k}}$, then the value of α is

• $- \frac{1}{2}$

• $\frac{1}{2}$

• 1

• − 1

The angle between velocity and acceleration of a particle describing uniform circular motion is

• 180°

• 90°

• 45°

• 60°

A very large number of balls are thrown vertically upwards in quick successions in such a way that the next ball is thrown when the previous one is at the maximum height. If the maximum height is 5 m, the number of balls thrown per minute is (take g = 10 m/s²)

• 80

• 120

• 40

• 60

A projectile is moving at 20 ms^-1 at its highest point, where it breaks into equal parts due to an internal explosion. One part moves vertically up at 30 ms^-1 with respect to the ground. Then the other part will move at

• 20 ms^-1

• $10\sqrt{31} {\mathrm{ms}}^{-1}$

• 50 ms^-1

• 30 ms^-1

100.

A body is projected vertically upwards from the surface of a planet of radius R with a velocity equal to half the escape velocity for that planet. The maximum height attained by the body is

• R/2

• R/3

• R/5

• R/4