A particle moving along x-axis has acceleration f, at time

A particle moving along x-axis has acceleration f, at time t, given by f = f₀(1 - (t/T)) , where f₀ and T are constants. The particle at t = 0 has zero velocity. In the time interval between t = 0 and the instant when f = 0, the particle’s velocity (v_x) is

Here, ƒ = ƒ₀		1 -	t		or,	dv	= ƒ₀		1 -	t
			T			dt				T

or, dv = ƒ₀		1 -	t		dt
			T

∴ v = ∫ dv = ∫		ƒ₀		1 -	t			dt
					T

or v = ƒ₀		t -	t²		+ C
			2T

where C is the constant of integration.
At t = 0, v = 0.

∴ 0 = ƒ₀		0 -	0		+ C ⇒ C = 0
			2T

∴ v = ƒ₀		t -	t²
			2T

if ƒ = 0, then

0 = ƒ₀		t -	t²		⇒ t = T
			2T

Hence, particle's velocity in the time interval t = 0 and t = T is given by

v_x = ^{t = T}∫_{t = 0} dv = ^T∫_{t = 0}		ƒ₀		1 -	t			dt
					T

= ƒ₀				t -	t²			^T
					2T			₀

= ƒ₀

T -

T²

= ƒ₀

T -

ƒ₀T