The motor exerts a force of F = (20t2) N on the cable, where t is in seconds. Determine the speed of the 25-kg crate when t = 4 s. The coefficients of static and kinetic friction between the crate and the plane are ms = 0.3 and mk = 0.25, respectively.

Force F = 20 t² N so force is variable.

friction on the crate = mgμ, μ is coefficient of kinetic friction, m is mass

Net force

= F - mgμ

= 20 t² - mgμ = m dv / dt (Newton's 2 nd law)

m dv = (20 t² - mgμ) dt

Integrating on both sides,

m v = 20 (t³ / 3) - mgμ t

Putting the values,

25 x v = 20 x 4³/3 - 25 x 9.8 x. 25 x 4

25 v = 426.67 - 245

25 v = 181.67

v = 7.27

v = 7.27 m / s