Question

Find the proper length of the rod, if at the moment $t_A$ the coordinate of the point $A$ is equal to $x_A$, and at the moment $t_B$ the coordinate of the point $B$ is equal to $x_B$;

Answer 1

Given: A rod $AB$

at t= $t_A$ $X_A$=$x_A$

at t=$t_B$ $X_B$=$x_B$

To Find: The Length of rod AB.

Formula used: $v=\frac{△x}{△t}$

$△x$ : difference in initial & final position of a point in rod.

$△t$ : difference in initial & final time.

Solution:Let us assume that length of rod AB=l ; and $t_B$ > $t_A$

Now, according to this question.

$△t=t_B-t_A$ $(if{t}_B>t_A)$--------------------(1)

Hence, Let us take the front end of the rod.

So, distance travelled by front end of the rod will be $△x$;

at $t=t_A$ $X_A$=$x_A$

$t=t_B$ $X_A=x_B+l$

Hence, $△X_A=x_B+l-x_A$--------(2)

So from the formula given:

$v=\frac{△x}{△t}$------(3)

Hence we will substitute

eq(1) & eq(2) in eq(3) we get:

$v=\frac{x_B+l-x_A}{t_B-t_A}$

So by solving the above equation:

$(t_B-t_A)v$ =$x_B+l-x_A$

$l=(t_B-t_A)v+x_A-x_B$