Question

Position of a particle moving along the X- axis varies with time as $x = 50t - 5{t^2}$ , where $t$ is time in seconds. Time during which particle move along positive x-direction
A. $t < 6$ s
B. $t > 5$ s
C. \[t < 5\] s
D. $t > 6$ s

Answer 1

Hint: We have to find the time during which the particle moves along a positive x-direction. We make use of the concept of instantaneous velocity and acceleration and then we find the time required for the motion. We will find velocity and acceleration and then substitute boundary conditions.

Complete step by step answer:
The position of a particle moving along the X- axis varies with time as $x = 50t - 5{t^2}$ .
$x = 50t - 5{t^2} - - - - - - - - - - (1)$
Let us calculate velocity and acceleration,
$v = \dfrac{{dx}}{{dt}} = \dfrac{d}{{dt}}(50t - 5{t^2})$
$\therefore v = \left( {50 - 10t} \right)m{s^{ - 1}} - - - - - - - - - (2)$
And $a = \dfrac{{dv}}{{dt}} = \dfrac{d}{{dt}}\left( {\dfrac{{dx}}{{dt}}} \right) = \dfrac{d}{{dt}}\left( {50 - 10t} \right)$
$\therefore a = - 10\,m{s^{ - 2}} - - - - - - - - - - (3)$
Now, using the boundary conditions as the particle has displacement and velocity zero initially.
From equation $(2)$ , we have
$0 = 50 - 10t$
$\therefore t = 5$ s
As the acceleration is negative, the particle moves in a straight line along +X direction during the time $t < 5$ s.

Hence, option C is correct.

Note: When the velocity of the body increases with time, acceleration is positive and when the velocity of the body decreases with time, then the acceleration is negative. Here, acceleration is negative and thus at time $t = 5$ , the velocity of the particle is positive and decreasing and hence the particle executes straight line motion for $t < 5$ s.