GATE Questions & Answers of Impulse and Momentum (Linear and Angular) and Energy Formulations, Collisions

A point mass is shot vertically up from ground level with a velocity of 4 m/s at time, $ t=0 $. It loses 20% of its impact velocity after each collision with the ground. Assuming that the acceleration due to gravity is 10 $ m/s^2 $ and that air resistance is negligible, the mass stops bouncing and comes to complete rest on the ground after a total time (in seconds) of

•  (A)  1
•  (B)  2
•  (C)  4
•  (D)  $ \infty $

A point mass of 100 kg is dropped onto a massless elastic bar (cross-sectional area = 100 mm², length = 1 m, Young's moduls = 100 GPa) from a height H of 10 mm as shown (Figure is not to scale). If g = 10 m/s², the maximum compression of the elastic bar is ____________mm.

A point mass having mass $ M $ is moving with a velocity $ V $ at an angle $ \theta $ to the wall as shown in the figure. The mass undergoes a perfectly elastic collision with the smooth wall and rebounds. The total change (final minus initial) in the momentum of the mass is

•  (A)  $-2MV\;\cos\;\theta\;\widehat j$
•  (B)  $2MV\;\sin\;\theta\;\widehat j$
•  (C)  $2MV\;\cos\;\theta\;\widehat j$
•  (D)  $-2MV\;\sin\;\theta\;\widehat j$

A mass of 2000 kg is currently being lowered at a velocity of 2 m/s from the drum as shown in the figure. The mass moment of inertia of the drum is $150\mathrm{kg}-\mathrm m^2$. On applying the brake, the mass is brought to rest in a distance of 0.5 m. The energy absorbed by the brake (in kJ) is __________

A system of particles in motion has mass center $G$ as shown in the figure. The particle $i$ has mass $m_i$ and its position with respect to a fixed point $O$ is given by the position vector $ {\boldsymbol r}_\mathbf i $. The position of the particle with respect to $G$ is given by the vector $ {\boldsymbol\rho}_\mathbf i $. The time rate of change of the angular momentum of the system of particles about $G$ is

(The quantity $ {\overset{\boldsymbol.\boldsymbol.}{\mathbf\rho}}_\mathbf i\boldsymbol\infty $ indicates second derivative of $ {\boldsymbol p}_\mathbf i $ with respect to time and likewise for $ \boldsymbol r\boldsymbol i$ ).

•  (A)  $ {\textstyle\sum_i}r_i\times m_i\overset{..}{{\boldsymbol p}_i\;} $
•  (B)  $ {\textstyle\sum_i}\rho_i\times m_i{\overset{\boldsymbol.\boldsymbol.}{\mathbf r}}_i $
•  (C)  $ {\textstyle\sum_i}r_i\times m_i{\overset{\boldsymbol.\boldsymbol.}{\mathbf r}}_i $
•  (D)  $ {\textstyle\sum_i}\rho_i\times m_i{\overset{..}{\boldsymbol p}}_i $

A ball of mass 0.1 kg, initially at rest, is dropped from height of 1 m. Ball hits the ground and bounces off the ground. Upon impact with the ground, the velocity reduces by 20% . The height (in m) to which the ball will rise is _____.

A small ball of mass 1 kg moving with a velocity of 12 m/s undergoes a direct central impact with a stationary ball of mass 2 kg. The impact is perfectly elastic. The speed (in m/s) of 2 kg mass ball after the impact will be____.

A mass m₁ of 100 kg travelling with a uniform velocity of 5 m/s along a line collides with a stationary mass m₂ of 1000 kg. After the collision, both the masses travel together with the same velocity. The coefficient of restitution is

•  (A) 0.6
•  (B) 0.1
•  (C) 0.01
•  (D) 0

The coefficient of restitution of a perfectly plastic impact is

•  (A) 0
•  (B) 1
•  (C) 2
•  (D) ∞

A stone with mass of 0.1 kg is catapulted as shown in the figure. The total force F_x (in N) exerted by the rubber band as a function of distance x (in m) is given by F_x = 300x². If the stone is displaced by 0.1m from the un-stretched position (x = 0) of the rubber band, the energy stored in the rubber band is

•  (A) 0.01 J
•  (B) 0.1 J
•  (C) 1 J
•  (D) 10 J

During inelastic collision of two particles, which one of the following is conserved?

•  (A) total linear momentum only
•  (B) total kinetic energy only
•  (C) both linear momentum and kinetic energy
•  (D) neither linear momentum nor kinetic energy