Momentum is always conserved. Equation for conservation of momentum: 

There is only one velocity on the right since the two astronauts grab onto each other, thus they move together at the same velocity. Solve.

The equation for momentum is:

To maintain conservation of momentum, a new state must have the same momentum as a previous state:

Since the rock and the bean bag move together after the collision, . And since the bean bag is initially stationary, 

Plug in known values and solve.

To solve this problem, we need to consider the change in the ball's momentum. To do so, we'll use the following equation.

Rearrange the above equation to solve for the average force.

To solve this problem we need to use the relationship between force and impulse, which is given by the following equation:

This equation represents that the rate of change of momentum with respect to time is equal to the net force that causes said change in momentum. Thus:

Note that Joe must have an initial velocity of  before he begins to apply the upwards Force that accelerates him upwards, therefore our equation simplifies to:

Solve for :

Say that, when we hit the ground, we have a velocity , which is predetermined by whatever happens before the impact. When we hit the ground you will experience a force for some time. This force will cause the acceleration that reduces our velocity to zero and gets us to stop. Note that, regardless of how much time it takes us to stop, the change in momentum (impulse) is fixed, since it directly depends on how much our velocity changes:

 (since we come to a stop)

Note that the initial momentum does not depend on the impact force nor on how much time it takes to stop. The initial momentum depends on the velocity we have when we first hit the ground. This velocity is given by whatever happened before we hit the ground, which no longer concerns us since we only care about what happens from the moment we first hit the ground till the moment we stop. Yes, the time that passes for you to stop is very small, but it is impossible for it to be zero. So we have that the change in momentum (impulse) is a constant:

, since  is predetermined. 

Remember that any change in momentum for a given mass occurs because its velocity changes. The velocity of the mass changes due to an acceleration and an acceleration is caused by a force. This gives us a relationship between force and impulse:

In our scenario,  would be the impact force that stops us and  the time it takes us to stop. From the equation above, it is easy to see that, since  is fixed, when  gets larger  gets smaller, and the other way around. Therefore, we bend our knees to effectively increase the time it takes us to stop. Thus, diminishing the impact force as to avoid hurting ourselves.

In order to solve this problem we need to use the relationship between force and impulse:

Since the ball is moving with a Velocity of , we have that

Note that the final velocity of the ball is zero since it comes to a stop. We want the force, , experienced by the person's hand to be less than or equal to the maximum impact force

Mathematically:

Use the impulse equation and solve for time:

Impulse is defined as the change in momentum. 

Where  is the impulse,  is final momentum and  is the initial momentum. Recall:

Where  is mass and  is velocity. 

Our bowling ball has a mass of 10kg, with initial velocity of  and final velocity of . 

Plug in these two values into the equation for impulse and solve.

Impulse is the change in momentum. So all that is needed for this problem is to solve for the change in momentum. 

Note!!!! momentum is a vector quantity. This means that we must accound for the change in the ball's direction. This can be done by defining one of the directions negative.

For convience, I'll define the initial velocity as negative. Plugging in numbers we get

.

We never accounted for the time of the collision. We would have needed to do this if the problem asked for the force the bat applied. 

To solve this problem, we will use conservation of momentum.  The initial momentum of the system must be equal to the final momentum of the system if no external forces act on it.  It is important to note the directions and signs  of the velocities.  From this information, we may write:

To solve this problem, we must first note that this is a collision problem.  Secondly, we must be careful with the signs of the velocities associated with each vehicle.  

Because of the parameters of the problem, it is easy to see that the total momentum of the system initially is equal to zero.  Therefore, the final momentum of the system must also be zero.