Question

7. We studied objects in motion with constant acceleration in clion, and deduced some formulse for oonstant-ncceleration motion that you have beem using for the previous problems If the ncceleration is not coastant, they do not apply, but we ean go back to the sume procedure we did in claes to find other formulne that ere valid. Suppose, that an object moves with a time-varying acceleration that starts at zero but that increases lineurly with time, ie. a-kt (a) What mast the dianeusions of k be? What units might it he mewaired in? (b) Wlhst is the displacement of this objeet from its initial position (ie honw far hes it movwed) as n Eiunction of time

Answer 1

(a) Since the dimension of $a$ is m/s² , $k$ must have the dimension m/s³

(b)

$\frac{dv}{dt}=kt$

$\int_u^v dv'=\int_0^t kt'\,dt'$

where I assumed an initial velocity to be u, and velocity at a time t, to be v=v(t). So,

$v=u+\frac{1}{2}kt^2$

$\frac{dx}{dt}=u+\frac{1}{2}kt^2$

$\int_{x_0}^x dx'=\int_0^t\left (u+\frac{1}{2}kt'^2 \right )\,dt'$

Thus, the displacement is

$x-x_0=S=ut+\frac{1}{6}kt^3$