Systems of particles, momentum and collisions

A baseball of mass $0.15\mathrm{kg}$ reaches the bat horizontally with speed $35m/s$ and is hit back along the same direction, in the opposite sense, at $45m/s$. The contact lasts $1.5\mathrm{ms}$. Compute the magnitude of the impulse received by the ball and the average force exerted by the bat, neglecting the weight during the contact.

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A body of mass $4.0\mathrm{kg}$, initially at rest on a smooth horizontal plane, is pushed by a horizontal force of fixed direction and magnitude $F(t)=6.0t$, with $F$ in newtons and $t$ in seconds, for $0\le t\le 2.0\mathrm{s}$. Compute the impulse of the force and the velocity of the body at time $t=2.0\mathrm{s}$.

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Three particles of masses $m_1=2.0\mathrm{kg}$, $m_2=3.0\mathrm{kg}$, $m_3=5.0\mathrm{kg}$ are located at $(1.0,0)$, $(0,2.0)$ and $(4.0,4.0)$ respectively, with coordinates in meters. Determine the center of mass of the system.

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A uniform rod of length $1.2\mathrm{m}$ and mass $2.0\mathrm{kg}$ carries, fixed at one end, a small sphere of mass $1.0\mathrm{kg}$ and negligible size. Determine the distance of the center of mass of the system from the free end of the rod.

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A rifle of mass $3.5\mathrm{kg}$, free to move, fires horizontally a bullet of mass $7.0\mathrm{g}$ that leaves with velocity $500m/s$. Compute the recoil velocity of the rifle.

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A person of mass $60\mathrm{kg}$ stands still on a raft of mass $240\mathrm{kg}$, at rest in calm water. The person walks in a straight line for $5.0\mathrm{m}$ relative to the raft and stops. Neglecting the horizontal resistance of the water, compute the displacement of the raft and that of the person relative to the water.

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A railcar of mass $3.0\cdot 10^4\mathrm{kg}$, moving at $2.0m/s$ on a horizontal track, collides with a railcar at rest of mass $2.0\cdot 10^4\mathrm{kg}$ and the two stay coupled. Neglecting friction, compute the velocity after coupling and the fraction of kinetic energy dissipated.

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A bullet of mass $5.0\mathrm{g}$, with horizontal velocity $400m/s$, embeds itself in a block of mass $1.0\mathrm{kg}$ hanging at rest from inextensible strings of negligible mass. Neglecting air, compute the velocity of the block immediately after the collision and the maximum height to which it rises.

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A cart of mass $3.0\mathrm{kg}$ with velocity $4.0m/s$ collides elastically, along a straight frictionless horizontal track, with a cart at rest of mass $1.0\mathrm{kg}$. Compute the velocities of the two carts after the collision.

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A ball is dropped from rest from a height of $2.0\mathrm{m}$ above a rigid floor and, after the bounce, rises back to $1.28\mathrm{m}$. Neglecting air, compute the coefficient of restitution of the collision and the fraction of kinetic energy dissipated in the bounce.

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On a smooth horizontal plane a puck of mass $0.20\mathrm{kg}$ with velocity $3.0\hat{ı}m/s$ collides with a puck of mass $0.30\mathrm{kg}$ with velocity $2.0\hat{ȷ}m/s$, and the two stick together. Compute the magnitude and direction of the common velocity after the collision.

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Two bodies of masses $4.0\mathrm{kg}$ and $1.0\mathrm{kg}$ move along the same line with velocities $2.0m/s$ and $-3.0m/s$. Compute the velocity of the center of mass, the kinetic energy in the center-of-mass frame and the maximum energy that a collision between the two bodies can dissipate.

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A rocket in space, in the absence of external forces, expels gas with velocity $3.0km/s$ relative to the rocket. Compute the ratio between initial mass and final mass needed to go from rest to $6.0km/s$ and the fraction of the initial mass made up of fuel.

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