Aparticle of mass m moves along a straight line with initial speed vi. a force of magnitude f pushes the particle a distance d along the direction of its motion.

part a: find vf, the final speed of the particle after it has traveled a distance d. express the final speed in terms of vi, m, f, and d.

part b: by what multiplicative factor rk does the initial kinetic energy increase, and by what multiplicative factor rw does the work done by the force increase (with respect to the case when the particle had a mass m)? if one of the quantities doubles, for instance, it would increase by a factor of 2. if a quantity stays the same, then the multiplicative factor would be 1. you should enter the two factors separated by a comma.

part c: the particle's change in speed over the distance d will be the change in speed when it had a mass equal to m.

part d: by what factor rk does the initial kinetic energy increase (with respect to the first situation, with mass m and speed vi), and by what factor rw does the work done by the force increase? again, enter the two factors, separated by a comma.

Amass m = 74 kg slides on a frictionless track that has a drop, followed by a loop-the-loop with radius r = 18.4 m and finally a flat straight section at the same height as the center of the loop (18.4 m off the ground). since the mass would not make it around the loop if released from the height of the top of the loop (do you know why? ) it must be released above the top of the loop-the-loop height. (assume the mass never leaves the smooth track at any point on its path.) 1. what is the minimum speed the block must have at the top of the loop to make it around the loop-the-loop without leaving the track? 2. what height above the ground must the mass begin to make it around the loop-the-loop? 3. if the mass has just enough speed to make it around the loop without leaving the track, what will its speed be at the bottom of the loop? 4. if the mass has just enough speed to make it around the loop without leaving the track, what is its speed at the final flat level (18.4 m off the ground)? 5. now a spring with spring constant k = 15600 n/m is used on the final flat surface to stop the mass. how far does the spring compress?