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In this example , we are given that a toy boat in a swimming pool is pulled by a boy,
by a string as shown . if boy pulls the string at a constant speed u , find the speed with
which boat will move in the string makes an angle, theta with the horizontal. We are given
that boy is pulling the rope with the speed u. The obviously due to this the boat will
be moving toward left with a velocity v-b. Now In this situation we are required to find
out the value of this speed with which boat is moving. So let us consider, x is the distance
of the boat from edge of the swimming pool , if h is the height , and say the total length
of the string is. L. In this situation, we can say if this distance is y, from here to
here. when the boy is pulling the rope we can directly write, that y is changing at
a rate of u, so in this situation we can directly write. V-b, as minus d-x by d-t, because with
the velocity of boat axis decreasing , you take negative sign, and again due to this
speed u, this y is decreasing , so u can be written as, minus of d-y by d-t. In this situation
if we find out in the solution if we find out the relation in x and y. That can be very
easily written . relation in x and y , this can be given as x square + h square = y square.
If we differentiate it with respect to time, will get, 2 x d-x by d-t, different ion of
h will be 0, this = 2 y into, d-y by d-t. Here 2 gets cancelled out . we can write x
into d x by d t can be written as v-b , = y into d-y by d-t can be written as u negative
sign on both sides will cancelled out. So velocity of boat can be written as u upon.
X into y. Now here y by x we can simply write as seck theta , so this will be u seck theta,
which will be the answer to this problem this is the speed with which the boat will be moving
, on the water . if the rope is pulled , the string is pulled with speed u.
Let us see an alternative method for the same problem . here , if the boy pulls the string
with the speed u, and say the boat is moving with velocity v-b. Again we can say as the
string length is constant . the speed with which boy is pulling . along the rope, and
the speed with which , this end of the rope is moving along the rope, or along the string,
must remain same. As the length is constant we can say in this direction the speed with
which the end of rope is moving along the length is v-b coz theta. So here we can directly
state along the length of the string, the velocity component has to remain same as the
string remains tight. Now for . string to remain tight , we can directly write u has
to be= . v-b coz theta . so in this situation, the speed with which the string is moving
and the speed with which the boat is moving , along the length of the string has to remain
same. So directly we can write v-b = , u seck theta . that will be the answer we are approaching
directly , using the concept of , length of the string to remain, same. So here also we
can write down a note. Just similar to that we have studied in previous example . this
for an. inextensible rope . or inextensible string . always. Velocity components . of
all. points. On string .along its length . remain same. Or equal. All the velocity components
when it along the length of string, must remain , equal . so be careful about this also.