The boy jumping off his skateboard and the carts colliding are both examples of the conversation of linear momentum (which is very much related to Newton's second law).
An object's momentum is its mass times velocity:
p = mv
The general equation for conservation of momentum is:
p₁ = p₂
Whenever there is an interaction between any number of objects, the total momentum before is the same as the total momentum after. For simplicity's sake we mostly use this to keep track of the momenta of two objects:
m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
We can use this equation to predict the outcomes of the situations described in your homework.
Note that v₁ and v₁' are used to represent the velocity of m₁ before and after the event.
Chris is standing on his skateboard, initially at rest. He then jumps backward, giving himself a velocity in a certain direction. We know that neither Chris nor his skateboard are moving before he jumps off, so the total momentum before the jump, and therefore the total momentum after the jump, must be 0. Chris's new momentum and his skateboard's new momentum must add to 0. That must mean that the skateboard is given a velocity opposite to the direction of Chris's velocity. This explanation may seem like a mouthful, so you can use the general equation of the conservation of momentum to mathematically describe the situation:
m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
m₁ and v₁ are Chris's mass and velocity.
m₂ and v₂ are his skateboard's mass and velocity.
Both Chris and his skateboard are initially at rest, so you can use v₁ = v₂ = 0 to simplify the equation.
m₁v₁' + m₂v₂' = 0
m₁v₁' = -m₂v₂'
This allows you to see mathematically that, whatever new momentum Chris has after jumping off, his skateboard must have the same magnitude of momentum (and therefore a nonzero velocity) in the opposite direction.
Cart A is moving and collides with Cart B, which is initially at rest. The carts have velcro pads attached to them, so it is expected that the carts will stick together after the collision.
Let's use the general equation of the conservation of momentum again:
m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
Let's choose m₁ and v₁ to represent Cart A's mass and velocity.
m₂ and v₂ represents Cart B's mass and velocity.
The carts will stick together after the collision, so v₁' = v₂', so we can simplify the equation by replacing these terms with a single term v'
m₁v₁ + m₂v₂ = m₁v' + m₂v'
m₁v₁ + m₂v₂ = (m₁+m₂)v'
We can further simplify the equation knowing that v₂, Cart B's velocity, is 0.
m₁v₁ = (m₁+m₂)v'
The simplified equation above allows you to see mathematically that v' is guaranteed to be smaller than v₁. This means that Cart A and Cart B will move together at a velocity slower than Cart A's velocity before the collision.
If the math is unconvincing, consider the following:
If you have a moving object, then somehow suddenly increase its mass, the object must move at a slower velocity to have the same momentum. This is essentially what is happening when the carts collide. Cart A is an object with some mass and some velocity. m₁+m₂ is Cart A colliding with Cart B to create a bigger object. This represents the object suddenly gaining some extra mass. Again, to keep the same momentum, the bigger object must have a slower velocity than what it originally had.
