Item 11 A spring-loaded toy gun exerts a variable force on a plastic ball as the spring expands. Consider a horizontal spring and a ball of mass m whose position when barely touching a fully expanded spring is x = 0. The ball is pushed to the left, compressing the spring. The spring force on the ball, when the ball is at position x (which is negative), can be written as (FSp)x=−kx , where k is called the spring constant. The minus sign is needed to make the x-component of the force positive. Suppose the ball is initially pushed to x0=−L, then released and shot to the right.

Part A

Use what you've learned in calculus to write the correct proof so that a_{x}= v_{x}\frac{dv_{}{_{x}}}{dx}

Part B

Find an expression, in terms of m, k, and L, for the speed of the ball as it comes off the spring x = 0.

a and b sometimes have a positive value

i think the answer is a and b

i wish you luck

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.

find the surface area of the sphere to the nearest square unit. use a calculator.

a. 254 in.2

b. 64 in.2

c. 127 in.2

d. 1,018 in.2