A projectile is moving at $20\,m/sec$ at its highest point where it breaks into two equal parts due to an internal explosion. One part moves vertically up at $30\,m/sec$ . Then the other part will move at ............. $\mathrm{m}/ \mathrm{s}$
A projectile is moving at $20\,m/sec$ at its highest point where it breaks into two equal parts due to an internal explosion. One part moves vertically up at $30\,m/sec$ . Then the other part will move at ............. $\mathrm{m}/ \mathrm{s}$
- A
$20$
- B
$10\sqrt {31}$
- C
$50$
- D
$30$
Similar Questions
A space craft of mass $M$ is moving with velocity $V$ and suddenly explodes into two pieces. A part of it of mass m becomes at rest, then the velocity of other part will be
A space craft of mass $M$ is moving with velocity $V$ and suddenly explodes into two pieces. A part of it of mass m becomes at rest, then the velocity of other part will be
Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass
$(a)$ Show $p = p _{t}^{\prime}+m_{t} V$
where $p$, is the momentum of the the particle (of mass $m$ ) and $p_{t}^{\prime \prime}=m_{t} v_{t}$,
Note $v_{t}$, is the velocity of the particle relative to the centre of mass. Also, prove using the definition of the centre of mass $\sum p _{t}^{\prime}=0$
$(b)$ Show $K=K^{\prime}+1 / 2 M V^{2}$
where $K$ is the total kinetic energy of the system of particles. $K^{\prime}$ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and $M V^{2} / 2$ is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system).
$(c)$ Show $L = L ^{\prime}+ R \times M V$
where $L ^{\prime}=\sum r _{t}^{\prime} \times p _{t}^{\prime}$ is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember $r _{t}^{\prime}= r _{t}- R$; rest of the notation is the standard notation used in the chapter. Note $L$ ' and $M R \times V$ can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.
$(d)$ Show $\frac{d L ^{\prime}}{d t}=\sum r _{t}^{\prime} \times \frac{d p ^{\prime}}{d t}$
Further, show that
$\frac{d L ^{\prime}}{d t}=\tau_{e x t}^{\prime}$
where $\tau_{c t t}^{\prime}$ is the sum of all external torques acting on the system about the centre of mass. (Hint: Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.)
Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass
$(a)$ Show $p = p _{t}^{\prime}+m_{t} V$
where $p$, is the momentum of the the particle (of mass $m$ ) and $p_{t}^{\prime \prime}=m_{t} v_{t}$,
Note $v_{t}$, is the velocity of the particle relative to the centre of mass. Also, prove using the definition of the centre of mass $\sum p _{t}^{\prime}=0$
$(b)$ Show $K=K^{\prime}+1 / 2 M V^{2}$
where $K$ is the total kinetic energy of the system of particles. $K^{\prime}$ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and $M V^{2} / 2$ is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system).
$(c)$ Show $L = L ^{\prime}+ R \times M V$
where $L ^{\prime}=\sum r _{t}^{\prime} \times p _{t}^{\prime}$ is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember $r _{t}^{\prime}= r _{t}- R$; rest of the notation is the standard notation used in the chapter. Note $L$ ' and $M R \times V$ can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.
$(d)$ Show $\frac{d L ^{\prime}}{d t}=\sum r _{t}^{\prime} \times \frac{d p ^{\prime}}{d t}$
Further, show that
$\frac{d L ^{\prime}}{d t}=\tau_{e x t}^{\prime}$
where $\tau_{c t t}^{\prime}$ is the sum of all external torques acting on the system about the centre of mass. (Hint: Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.)
A person trying to lose weight (dieter) lifts a $10\; kg$ mass, one thousand times, to a hetght of $0.5\; m$ each time. Assume that the potential energy lost each time she lowers the mass is dissipated.
$(a)$ How much work does she do against the gravitational force?
$(b)$ Fat supplies $3.8 \times 10^{7} \;J$ of energy per kilogram which is converted to mechanical energy with a $20 \%$ efficiency rate. How much fat will the dieter use up?
A person trying to lose weight (dieter) lifts a $10\; kg$ mass, one thousand times, to a hetght of $0.5\; m$ each time. Assume that the potential energy lost each time she lowers the mass is dissipated.
$(a)$ How much work does she do against the gravitational force?
$(b)$ Fat supplies $3.8 \times 10^{7} \;J$ of energy per kilogram which is converted to mechanical energy with a $20 \%$ efficiency rate. How much fat will the dieter use up?
A particle of mass $m$ moving horizontally with $v_0$ strikes $a$ smooth wedge of mass $M$, as shown in figure. After collision, the ball starts moving up the inclined face of the wedge and rises to $a$ height $h$. When the particle has risen to $a$ height $h$ on the wedge, then choose the correct alternative $(s)$
A particle of mass $m$ moving horizontally with $v_0$ strikes $a$ smooth wedge of mass $M$, as shown in figure. After collision, the ball starts moving up the inclined face of the wedge and rises to $a$ height $h$. When the particle has risen to $a$ height $h$ on the wedge, then choose the correct alternative $(s)$
In an elastic collision of two billiard balls, which of the following quantities remain conserved during the short time of collision of the balls ? (i.e. when they are in contact)
$(a)$ Kinetic energy.
$(b)$ Total linear momentum.
Give reason for your answer in each case.
In an elastic collision of two billiard balls, which of the following quantities remain conserved during the short time of collision of the balls ? (i.e. when they are in contact)
$(a)$ Kinetic energy.
$(b)$ Total linear momentum.
Give reason for your answer in each case.