Advertisements
Advertisements
Question
The blocks shown in figure have equal masses. The surface of A is smooth but that of Bhas a friction coefficient of 0.10 with the floor. Block A is moving at a speed of 10 m/s towards B which is kept at rest. Find the distance travelled by B if (a) the collision is perfectly elastic and (b) the collision is perfectly inelastic.
Advertisements
Solution
Given,
Speed of the block A = 10 m/s
The block B is kept at rest.
Coefficient of friction between the floor and block B, μ = 0.10
Let v1 and v2 be the velocities of A and B after collision, respectively.
(a) If the collision is perfectly elastic, linear momentum is conserved.
Using the law of conservation of linear momentum, we can write:
\[m u_1 + m u_2 = m v_1 + m v_2 \]
\[ \Rightarrow 10 + 0 = v_1 + v_2 \]
\[ v_1 + v_2 = 10 . . . \left( 1 \right)\]
\[\text{ We know,} \]
\[\text{ Velocity of separation } \left( \text{ after collision }\right) = \text{ Velocity of approach } \left( \text{ before collision } \right )\]
\[ v_1 - v_2 = - ( u_1 - v_2 )\]
\[ \Rightarrow v_1 - v_2 = - 10 . . . \left( 2 \right)\]
\[\text{ Substracting equation (2) from (1), we get: }\]
\[2 v_2 = 20\]
\[ \Rightarrow v_2 = 10 \text{ m/s }\]
The deceleration of block B is calculated as follows:
Applying the work-energy principle, we get:
\[\left( \frac{1}{2} \right) \times m \times (0 )^2 - \left( \frac{1}{2} \right) \times m \times v^2 = - m \times a \times s_1 \]
\[ \Rightarrow - \left( \frac{1}{2} \right) \times (10 )^2 = - \mu g \times s_1 \]
\[ \Rightarrow s_1 = \frac{100}{2 \times 0.1 \times 10} = 50 \text{ m }\]
(b) If the collision is perfectly inelastic, we can write:
\[m \times u_1 + m \times u_2 = (m + m) \times v\]
\[ \Rightarrow m \times 10 + m \times 0 = 2m \times v\]
\[ \Rightarrow v = \left( \frac{10}{2} \right) = 5 \text{ m/s }\]
The two blocks move together, sticking to each other.
∴ Applying the work-energy principle again, we get:
\[\left( \frac{1}{2} \right) \times 2m \times (0 )^2 - \left( \frac{1}{2} \right) \times 2m \times (v )^2 = 2m \times \mu g \times s_2 \]
\[ \Rightarrow \frac{(5 )^2}{0 . 1 \times 10 \times 2} = s_2 \]
\[ \Rightarrow s_2 = 12 . 5\]m
APPEARS IN
RELATED QUESTIONS
A bob suspended from the ceiling of a car which is accelerating on a horizontal road. The bob stays at rest with respect to the car with the string making an angle θ with the vertical. The linear momentum of the bob as seen from the road is increasing with time. Is it a violation of conservation of linear momentum? If not, where is the external force changes the linear momentum?
If the total mechanical energy of a particle is zero, is its linear momentum necessarily zero? Is it necessarily nonzero?
Internal forces can change
A man of mass 50 kg starts moving on the earth and acquires a speed 1.8 m/s. With what speed does the earth recoil? Mass of earth = 6 × 1024 kg.
A neutron initially at rest, decays into a proton, an electron, and an antineutrino. The ejected electron has a momentum of 1.4 × 10−26 kg-m/s and the antineutrino 6.4 × 10−27kg-m/s.
Find the recoil speed of the proton
(a) if the electron and the antineutrino are ejected along the same direction and
(b) if they are ejected along perpendicular directions. Mass of the proton = 1.67 × 10−27 kg.
A man of mass M having a bag of mass m slips from the roof of a tall building of height H and starts falling vertically in the following figure. When at a height h from the ground, the notices that the ground below him is pretty hard, but there is a pond at a horizontal distance x from the line of fall. In order to save himself he throws the bag horizontally (with respect to himself) in the direction opposite to the pond. Calculate the minimum horizontal velocity imparted to the bag so that the man lands in the water. If the man just succeeds to avoid the hard ground, where will the bag land?
A ball of mass 50 g moving at a speed of 2.0 m/s strikes a plane surface at an angle of incidence 45°. The ball is reflected by the plane at equal angle of reflection with the same speed. Calculate (a) the magnitude of the change in momentum of the ball (b) the change in the magnitude of the momentum of the ball.
Light in certain cases may be considered as a stream of particles called photons. Each photon has a linear momentum h/λ where h is the Planck's constant and λ is the wavelength of the light. A beam of light of wavelength λ is incident on a plane mirror at an angle of incidence θ. Calculate the change in the linear momentum of a photon as the beam is reflected by the mirror.
In a typical Indian Bugghi (a luxury cart drawn by horses), a wooden plate is fixed on the rear on which one person can sit. A bugghi of mass 200 kg is moving at a speed of 10 km/h. As it overtakes a school boy walking at a speed of 4 km/h, the boy sits on the wooden plate. If the mass of the boy is 25 kg, what will be the plate. If the mass of the boy is 25 kg, what will be the new velocity of the bugghi ?
A ball of mass m moving at a speed v makes a head-on collision with an identical ball at rest. The kinetic energy of the balls after the collision is three fourths of the original. Find the coefficient of restitution.
A block of mass 2.0 kg is moving on a frictionless horizontal surface with a velocity of 1.0 m/s (In the following figure) towards another block of equal mass kept at rest. The spring constant of the spring fixed at one end is 100 N/m. Find the maximum compression of the spring.
A bullet of mass 20 g travelling horizontally with a speed of 500 m/s passes through a wooden block of mass 10.0 kg initially at rest on a level surface. The bullet emerges with a speed of 100 m/s and the block slides 20 cm on the surface before coming to rest. Find the friction coefficient between the block and the surface (See figure).
A bullet of mass 10 g moving horizontally at a speed of 50√7 m/s strikes a block of mass 490 g kept on a frictionless track as shown in figure. The bullet remains inside the block and the system proceeds towards the semicircular track of radius 0.2 m. Where will the block strike the horizontal part after leaving the semicircular track?
The friction coefficient between the horizontal surface and each of the block shown in figure is 0.20. The collision between the blocks is perfectly elastic. Find the separation between the two blocks when they come to rest. Take g = 10 m/s2.
Suppose the particle of the previous problem has a mass m and a speed \[\nu\] before the collision and it sticks to the rod after the collision. The rod has a mass M. (a) Find the velocity of the centre of mass C of the system constituting "the rod plus the particle". (b) Find the velocity of the particle with respect to C before the collision. (c) Find the velocity of the rod with respect to C before the collision. (d) Find the angular momentum of the particle and of the rod about the centre of mass C before the collision. (e) Find the moment of inertia of the system about the vertical axis through the centre of mass C after the collision. (f) Find the velocity of the centre of mass C and the angular velocity of the system about the centre of mass after the collision.
A metre stick is held vertically with one end on a rough horizontal floor. It is gently allowed to fall on the floor. Assuming that the end at the floor does not slip, find the angular speed of the rod when it hits the floor.
A uniform rod pivoted at its upper end hangs vertically. It is displaced through an angle of 60° and then released. Find the magnitude of the force acting on a particle of mass dm at the tip of the rod when the rod makes an angle of 37° with the vertical.
A thin spherical shell of radius R lying on a rough horizontal surface is hit sharply and horizontally by a cue. Where should it be hit so that the shell does not slip on the surface?
The track shown is figure is frictionless. The block B of mass 2m is lying at rest and the block A or mass m is pushed along the track with some speed. The collision between Aand B is perfectly elastic. With what velocity should the block A be started to get the sleeping man awakened?
