Bob B of mass $m$ at rest is hanging vertically from the ceiling via a massless string of length $10\text{ m}$, as shown in the figure. Point mass A of mass $m$ travelling horizontally with speed $10\text{ ms}^{-1}$ hits bob B elastically. The bob B rises $h$ meter after the collision. Taking the acceleration due to gravity $g = 10\text{ ms}^{-2}$ and neglecting the size of the bob, the value of $h$ is :
To find the height \(h\) to which bob B rises after an elastic collision, we need to use principles of conservation of momentum and energy.
- Since the collision is elastic, both momentum and kinetic energy are conserved.
- Before the collision:
- Momentum of A: \(p_A = m \times 10\ \text{ms}^{-1}\)
- Momentum of B: \(p_B = 0\) (since B is at rest)
- Total initial momentum: \(m \times 10\ \text{ms}^{-1}\)
- Let \(v_A\) and \(v_B\) be the velocities of A and B after collision, respectively. Using momentum conservation:
m \times 10 = m \times v_A + m \times v_B
- Using kinetic energy conservation:
\frac{1}{2} \times m \times 10^2 = \frac{1}{2} \times m \times v_A^2 + \frac{1}{2} \times m \times v_B^2
- Solving the equations, we find:
- \(v_A = 0\) (A comes to rest)
- \(v_B = 10\ \text{ms}^{-1}\)
- After the collision, bob B moves vertically upwards. Using energy conservation for B:
- Initial kinetic energy of B: \(\frac{1}{2} \times m \times (10)^2\)
- At the highest point, all kinetic energy converts to potential energy: \(mgh\)
- Set kinetic energy equal to potential energy:
\frac{1}{2} \times m \times (10)^2 = m \times g \times h
h = \frac{50}{10} = 5\ \text{m}
- Therefore, the height \(h\) is 5 meters.
The correct answer is 5.
Similar Questions
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- The amount of work done to raise a mass 'm' from the surface of the Earth to a height equal to the radius of the Earth 'R', will be :
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- When a ruler falls vertically, 5 different persons catch it with different reaction times. ($g = 9.8 \text{ m s}^{-2}$)
A. Person A has reaction time of $0.20 \text{ s}$.
B. Person B has reaction time of $0.22 \text{ s}$.
C. Person C has reaction time of $0.18 \text{ s}$.
D. Person D has reaction time of $0.19 \text{ s}$.
E. Person E has reaction time of $0.21 \text{ s}$.
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Important Questions from Mechanics
- When both jaws of vernier callipers touch each other, zero mark of the vernier scale is right to zero mark of main scale, $4^{\text{th}}$ mark on vernier scale coincides with certain mark on the main scale. While measuring the length of a cylinder, observer observes $15$ divisions on main scale and $5^{\text{th}}$ division of vernier scale coincides with a main scale division. Measured length of cylinder is ________mm.
(Least count of Vernier calliper = $0.1 \text{ mm}$) - Water drops fall from a tap on the floor, $5 \text{ m}$ below, at regular intervals of time, the first drop strikes the floor when the sixth drop begins to fall. The height at which the fourth drop will be from ground, at the instant when the first drop strikes the ground is ________m.
($g = 10 \text{ m/s}^2$) - Two wires $A$ and $B$ made of different materials of lengths $6.0 \text{ cm}$ and $5.4 \text{ cm}$, respectively and area of cross sections $3.0 \times 10^{-5} \text{ m}^2$ and $4.5 \times 10^{-5} \text{ m}^2$, respectively are stretched by the same magnitude under a given load. The ratio of the Young's modulus of $A$ to that of $B$ is $x : 3$. The value of $x$ is________.
Two circular discs of radius each $10 \text{ cm}$ are joined at their centres by a rod of length $30 \text{ cm}$ and mass $600 \text{ gm}$ as shown in figure.
If the mass of each disc is $600 \text{ gm}$ and applied torque between two discs is $43 \times 10^5 \text{ dyne.cm}$, the angular acceleration of the discs about the given axis $AB$ is________$\text{rad/s}^2$.
- A particle of mass $m$ falls from rest through a resistive medium having resistive force, $F = -kv$, where $v$ is the velocity of the particle and $k$ is a constant. Which of the following graphs represents velocity ($v$) versus time ($t$)?
