If someone is riding in the back of a pickup truck and throws a softball straight backward, is it possible for the ball to fall straight down as viewed by a person standing at the side of the road? under what condition would this occur? how would the motion of the ball appear to the person who threw it?

The person on the truck would have the same speed as that of the truck. Also, before the ball is thrown, the ball would have the speed as that of the truck.

It is possible for a person standing at the side of the road to view the falling straight down.  This would be possible if the ball is thrown backwards with the same magnitude of velocity as that of truck.

To the person on the truck, the ball would appear to have forward motion with the speed of the truck.

initial speed when it is dropped is 150 ft/s

now at the same moment the acceleration due to gravity is perpendicular to its velocity

so as we know that

now we have centripetal acceleration a = 32 ft/s^2 as its perpendicular to velocity

while tangential acceleration at the same moment will be zero

so

now radius of curvature is given as

so the radius of curvature just after it is dropped is 703.125 ft