Question

A body is initially at rest. It undergoes one dimensional motion with constant acceleration, the power delivered to it at time t is proportional to:
(A). ${t^{\dfrac{1}{2}}}$
(B). $t$
(C). ${t^{\dfrac{3}{2}}}$
(D). ${t^2}$

Answer 1

Hint: One dimension, one means movement along a line, or in a specific direction. Think of a car running down a straight line, or someone running on a straight route. You can also think about an object being thrown up in the air vertically and watching it fall.
Formula used: $s = ut + \dfrac{1}{2}a{t^2}$

Complete step-by-step solution -
A body undergoes one dimensional motion with constant acceleration, in accordance with the question. So, in time it will cover a distance "t" is $s = ut + \dfrac{1}{2}a{t^2}$
It is initially provided that the body is at rest So, $u = 0$
Now,
$s = \dfrac{1}{2}a{t^2}$ where a is constant
Power supplied is $P = \dfrac{{work\;done}}{{time}} = \dfrac{w}{t} = \dfrac{{F \times S}}{t}$
$ = \dfrac{{F \times S}}{T} = \dfrac{{\left( {ma} \right) \times \left( {\dfrac{1}{2}a{t^2}} \right)}}{t} = \left( {\dfrac{1}{2}m{a^2}} \right)t$
$P \propto t$
Therefore, the delivered power is proportional to "t." Hence the option B is correct.

Note: In daily life, movement with constant acceleration occurs if an object is dropped: the body travels downward under the influence of gravity with the constant acceleration \[9.81{\text{ m}}{{\text{s}}^{ - 2}}\].
Acceleration is the rate of change in an object's velocity in relation to time. Accelerations are quantities of the vectors (in that they are of magnitude and direction). The acceleration orientation of an object is given by the orientation of the net force acting upon that object. The magnitude of the acceleration of an object, as Newton's Second Law defines it, is the combined effect of two causes:
i). The net strength of all external forces acting upon that object — magnitude is directly proportional to the resulting net force;
ii). The mass of the object, depending on the materials it is made from — the value is inversely proportional to the mass of the object.