6. A rocket burning its onboard fuel while moving through space has velocity −→v (t) and mass m(t) at time t. If the exhaust gases escape with velocity −→v e, relative to the rocket, then md−→v /dt = dm/dt−→v e. (a) Show using this equation that −→v (t) = −→v (0) − ln(m(0)/m(t))−→v e; (b) If the rocket accelerates in a straight line from rest to twice the speed of its exhaust gases, then what fraction of its initial mass would the rocket have to burn as fuel?

Astudent throws a water balloon with speed v0 from a height h = 1.76 m at an angle θ = 21° above the horizontal toward a target on the ground. the target is located a horizontal distance d = 9.5 m from the student’s feet. assume that the balloon moves without air resistance. use a cartesian coordinate system with the origin at the balloon's initial position. (a) what is the position vector, rtarge t, that originates from the balloon's original position and terminates at the target? put this in terms of h and d, and represent it as a vector using i and j. (b) in terms of the variables in the problem, determine the time, t, after the launch it takes the balloon to reach the target. your answer should not include h. (c) create an expression for the balloon's vertical position as a function of time, y(t), in terms of t, vo, g, and θ. (d) determine the magnitude of the balloon's initial velocity, v0, in meters per second, by eliminating t from the previous two expressions.

Calculate the first and second velocities of the car with four washers attached to the pulley, using the formulas v1 = 0.25 m / t1 , and v2 = 0.25 m / (t2 – t1) where t1 and t2 are the average times the car took to reach the 0.25 and the 0.50 meter marks. record these velocities, to two decimal places, in table e.

The direction of the current in an alternating current circuit remains constant. is direct. changes repeatedly. changes randomly.