A box of mass m sliding with an initial speed of v on a rough horizontal surface runs into a fixed spring of elastic constant k, compressing it a distance x1 = x from its relaxed position while momentarily coming to rest. There is NO FRICTION. What would be the right equation for energy balance in this example ?

Attach the free body diagram and i will you answer the question.! : d

Since you have not included the experiment situation and the data, of course, i can not give you specific details about it. but i can you with this question.1) i will assume that the motion is projectile (parabolic) motion.2) yes, the horizontal distance depends on the horizontal velocity as, naming vx the horizontal velocity and t the time, it is: x = vx × t3) now, observe that the flight time depends on the vertical velocity as, naming voy the initial vertical velocity, t the time and y the vertical distance before reaching the ground: y = yo + voy×t - gt² / 2.the higher the initial vertical velocity (voy) the longer the object will be flying and so t will be higher in the equation for the horizontal distance.that explains the relationship between the horizontal distance and the horizontal velocity.

Answervery cold;