Advertisements
Advertisements
प्रश्न
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?
Advertisements
उत्तर
Mass of man = M
Initial velocity of the man = 0
Mass of bag = m
Let the man throws the bag towards left with a velocity v and himself moves towards right with a velocity V.
Using the law of conservation of momentum,
\[mv = MV\]
\[ \Rightarrow v = \frac{MV}{m} . . . (1)\]
\[\text{ Let the total time he takes to reach ground be t_1 . }\]
\[ \Rightarrow t_1 = \sqrt{\frac{2H}{g}}\]
\[\text{ Let the total time he takes to reach the height h be t_2 .} \]
\[ \Rightarrow t_2 = \sqrt{\frac{2(H - h)}{g}}\]
\[ \therefore \text{ The time of flying in covering the remaining height h is,} \]
\[ t = t_1 - t_2 \]
\[ \Rightarrow t = \sqrt{\frac{2H}{g}} - \sqrt{\frac{2(H - h)}{g}}\]
\[ = \sqrt{\frac{2}{g}}\left( \sqrt{H} - \sqrt{H - h} \right)\]
During this time, the man covers a horizontal distance x and lands in the water.
\[\Rightarrow x = V \times t\]
\[ \Rightarrow V = \frac{x}{t}\]
\[ \therefore v = \frac{M}{m}\frac{x}{t} \left[ \text{ using equation } \left( 1 \right) \right]\]
\[ = \frac{M}{m}\frac{x}{\sqrt{\frac{2}{g}}\left( \sqrt{H} - \sqrt{H - h} \right)}\]
Thus, the minimum horizontal velocity imparted to the bag, such that the man lands in the water is
\[\frac{M}{m}\frac{x}{\sqrt{\frac{2}{g}}\left( \sqrt{H} - \sqrt{H - h} \right)}\]
Let the bag lands at a distance x' towards left from actual line of fall.
As there is no external force in horizontal direction, the x-coordinate of the centre of mass will remain same.
\[\Rightarrow 0 = \frac{M \times (x) + m \times x'}{M + m}\]
\[ \Rightarrow x' = - \frac{M}{m}x\]
Therefore, the bag will land at a distance
\[\frac{M}{m}x\]
APPEARS IN
संबंधित प्रश्न
If the total mechanical energy of a particle is zero, is its linear momentum necessarily zero? Is it necessarily nonzero?
Use the definition of linear momentum from the previous question. Can we state the principle of conservation of linear momentum for a single particle?
When a nucleus at rest emits a beta particle, it is found that the velocities of the recoiling nucleus and the beta particle are not along the same straight line. How can this be possible in view of the principle of conservation of momentum?
The quantities remaining constant in a collisions are
A shell is fired from a cannon with a velocity V at an angle θ with the horizontal direction. At the highest point in its path, it explodes into two pieces of equal masses. One of the pieces retraces its path to the cannon. The speed of the other piece immediately after the explosion is
In an elastic collision
(a) the kinetic energy remains constant
(b) the linear momentum remains constant
(c) the final kinetic energy is equal to the initial kinetic energy
(d) the final linear momentum is equal to the initial linear momentum.
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.
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.
A gun is mounted on a railroad car. The mass of the car, the gun, the shells and the operator is 50 m where m is the mass of one shell. If the velocity of the shell with respect to the gun (in its state before firing) is 200 m/s, what is the recoil speed of the car after the second shot? Neglect friction.
A 60 kg man skating with a speed of 10 m/s collides with a 40 kg skater at rest and they cling to each other. Find the loss of kinetic energy during the collision.
Consider a head-on collision between two particles of masses m1 and m2. The initial speeds of the particles are u1 and u2 in the same direction. the collision starts at t = 0 and the particles interact for a time interval ∆t. During the collision, the speed of the first particle varies as \[v(t) = u_1 + \frac{t}{∆ t}( v_1 - u_1 )\]
Find the speed of the second particle as a function of time during the collision.
A block of mass 200 g is suspended through a vertical spring. The spring is stretched by 1.0 cm when the block is in equilibrium. A particle of mass 120 g is dropped on the block from a height of 45 cm. The particle sticks to the block after the impact. Find the maximum extension of the spring. Take g = 10 m/s2.
A bullet of mass 25 g is fired horizontally into a ballistic pendulum of mass 5.0 kg and gets embedded in it. If the centre of the pendulum rises by a distance of 10 cm, find the speed of the bullet.
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?
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 small disc is set rolling with a speed \[\nu\] on the horizontal part of the track of the previous problem from right to left. To what height will it climb up the curved part?
A sphere starts rolling down an incline of inclination θ. Find the speed of its centre when it has covered a distance l.
The following figure shows a small spherical ball of mass m rolling down the loop track. The ball is released on the linear portion at a vertical height H from the lowest point. The circular part shown has a radius R.
(a) Find the kinetic energy of the ball when it is at a point A where the radius makes an angle θ with the horizontal.
(b) Find the radial and the tangential accelerations of the centre when the ball is at A.
(c) Find the normal force and the frictional force acting on the if ball if H = 60 cm, R = 10 cm, θ = 0 and m = 70 g.
