In a rocket-propulsion problem the mass is variable. another such problem is a raindrop falling through a cloud of small water droplets. some of these small droplets adhere to the raindrop, thereby increasing its mass as it falls. the force on the raindrop is
fext=dpdt=mdvdt+vdmdt
suppose the mass of the raindrop depends on the distance x that it has fallen. then m=kx, wherek is a constant, and dm/dt=kv. this gives, since fext=mg,
mg=mdvdt+v(kv)
or, dividing by k,
xg=xdvdt+v2
this is a differential equation that has a solution of the form v=at, where a is the acceleration and is constant. take the initial velocity of the raindrop to be zero.
a) using the proposed solution for v, find the acceleration a.
b) find the distance the raindrop has fallen in
t = 2.00s .
c) given that
k = 2.00g/m , find the mass of the raindrop at t = 2.00s

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Search coils and credit cards. one practical way to measure magnetic field strength uses a small, closely wound coil called a search coil. the coil is initially held with its plane perpendicular to a magnetic field. the coil is then either quickly rotated a quarter-turn about a diameter or quickly pulled out of the field. (a) derive the equation relating the total charge q that flows through a search coil to the magnetic-field magnitude b. the search coil has n turns, each with area a, and the flux through the coil is decreased from its initial maximum value to zero in a time ∆t. the resistance of the coil is r, and the total charge is q = i∆t, where i is the average current induced by the change in flux. (b) in a credit card reader, the magnetic strip on the back of a credit card is rapidly “swiped” past a coil within the reader. explain, using the same ideas that underlie the operation of a search coil, how the reader can decode the information stored in the pattern of magnetization on the strip. (c) is it necessary that the credit card be “swiped” through the reader at exactly the right speed? why or why not?

Two manned satellites approaching one another at a relative speed of 0.550 m/s intend to dock. the first has a mass of 2.50 ✕ 103 kg, and the second a mass of 7.50 ✕ 103 kg. assume that the positive direction is directed from the second satellite towards the first satellite. (a) calculate the final velocity after docking, in the frame of reference in which the first satellite was originally at rest.(b) what is the loss of kinetic energy in this inelastic collision? (c) repeat both parts, in the frame of reference in which the second satellite was originally at rest. final velocity(d) loss of kinetic energy = ?