Question

A particle moves in the $xy$ plane with a constant acceleration $g$ in the negative $y$-direction. Its equation of motion is $y=ax-b{x}^2$, where $a$ and $b$ are constants. Which of the following are correct?

• A. The $x$-component of its velocity is constant
• B. At the origin, the y-component of its velocity is $a\sqrt{\frac{g}{2b}}$
• C. The particle moves exactly like a projectile
• D. None of these

Answer 1

Answer: (A), (B), (C)

The correct options are
A The $x$-component of its velocity is constant
B At the origin, the y-component of its velocity is $a\sqrt{\frac{g}{2b}}$
C The particle moves exactly like a projectile

As in this it is given that the particle will only have the acceleration in the negative y- direction so the acceleration in the x - direction is zero

so $a_x=0$

so the velocity in the x-direction will be constant as acceleration in the x - direction is zero

now $y=ax-b{x}^2$

so now differentiate the equation so we get

now differentiate this again

that is

as $a_x=0$ and $a_y=-g$

so

so

so as we know

as at origin means x =0

so

and the graph or trajectory of is like parabola that like projectile as shown above

so option A , B , C are correct