Question

Two blocks of masses m, and my are kept on a smooth horizontal surface. A spring of mass m and natural leng Lo connects the two blocks as shown in the figure. +=0, m, and my are given velocities v, and 2v as show Just after t = 0, what is the value of x, so that the point remains at rest?

Answer 1

$\displaystyle x=\frac{L_0}{3}$

Answer 2

We choose a coordinate along the unstretched spring of natural length $L_0$ with $x=0$ at the left end (attached to the block of mass $m$ given velocity $v$) and $x=L_0$ at the right end (attached to the other block of mass $m_y$ given speed $2v$). (In problems of this type the directions are taken so that one velocity is positive and the other negative. Here we let the left mass move to the right with speed $v$ and the right mass move to the left with speed $2v$.)

Immediately after the impulse the spring (having non‐zero mass $m$) does not “know” about the impulse at the other end. Thus, while the ends instantaneously acquire the velocities of the blocks, the velocity of points in the spring must vary continuously from one end to the other. (In many elementary texts this is taken as a linear variation along the spring.)

So assume that a point at distance $x$ (measured from the left end) has velocity

$$v(x)=v +\left[\frac{v_R - v_L}{L_0}\right]\,x,$$

with

$$v_L=v\quad\text{and}\quad v_R=(-2v)$$

(the minus sign comes because the right–end velocity is opposite to the left–end’s direction). Thus

$$v(x)=v +\left[\frac{-2v-v}{L_0}\right]x=v - \frac{3v}{L_0}\,x.$$

We now require that there be a point on the spring which is at rest (i.e. $v(x)=0$). Setting

$$v-\frac{3v}{L_0}\,x=0,$$

we solve for $x$:

$$\frac{3v}{L_0}\,x=v\quad\Longrightarrow\quad x=\frac{L_0}{3}.$$

Thus the point located at a distance $\frac{L_0}{3}$ from the left end (attached to mass $m$) remains at rest instantaneously.

Core Explanation:

Assume a linear velocity distribution along the spring from $v$ at $x=0$ to $-2v$ at $x=L_0$. Write

$$v(x)=v-\frac{3v}{L_0}\,x.$$

Setting $v(x)=0$ gives $x=L_0/3$.