Which situation will produce the greatest change of momentum for a 1.0-kilogram cart?

Momentum is the product of mass of an object or a body and the velocity. According to second newtons' law of motion the resultant force is directly proportional to the rate of change in momentum.
Therefore, F = (mv-mu)/t where m is the mass, v- final velocity and u- initial velocity. 
Change in momentum is mv-mu, which can be given by Ft (impulse) from the law. 
Therefore in this case, the change in momentum will be higher when a net force of 5 N is applied for 2 seconds. ( change in momentum = (5×2) = 10 )