Newton's second law states that the resultant of the forces applied to an object is equal to the product between the object's mass and its acceleration:
[tex]\sum F = ma[/tex]
where in our problem, m is the mass the (child+cart) and a is the acceleration of the system.
We are only concerned about what it happens on the horizontal axis, so there are two forces acting on the cart+child system: the force F of the man pushing it, and the frictional force [tex]F_f[/tex] acting in the opposite direction. So Newton's second law can be rewritten as
[tex]F-F_a = ma[/tex]
or
[tex]F=ma + F_f[/tex]
since the frictional force is 15 N and we want to achieve an acceleration of [tex]a=1.50 m/s^2[/tex], we can substitute these values to find what is the force the man needs:
[tex]F=(30 kg)(1.5 m/s^2)+15 N=60 N[/tex]
