In this question, we're told that an object of a given mass is free-falling in the air with no air resistance. We're asked to determine the amount of upward force necessary to cause this object's velocity to become constant.

In order for an object's velocity to be maintained at a constant value, that object must not be accelerating. Therefore, we'll need to determine what force will allow this object to have a net acceleration of zero.

Looking at the falling object's motion in the downward direction, we know that it is falling due to the influence of gravity. Since we know its mass, we can calculate this downward force:

We've stated that in order to have a constant velocity, we need to have a net acceleration of zero. And since we know that the object is accelerating by  in the downward direction, we will need it to accelerate upward by  to cancel it to zero.

To conclude, we'll have to match the downward force by presenting an equal upward force in order to cancel out the object's acceleration and bring its velocity to a constant value.

Imagine two men of equal strength pulling on ropes that are connected to a crate. In order to obtain the largest resulting force the two men should pull in the same direction at . Pulling on opposite ends at  would result in zero resultant force and anything besides  would cause the force to act in an unneeded direction.

Using definition of force and superposition of Forces

Each person is apply the same amount of force against gravity. The car is holding still so it has no acceleration.

Solving for and plugging in values, remembering that gravity points downwards and thus is negative.

Using

Converting to and plugging in values.

In the question stem, we're told that two forces of equal magnitude are pointing south and the other east. We're then asked to find which way the third force must be oriented in order to cancel out the other two forces. To figure this out, it's best to draw a force diagram.

Looking at this force diagram, we actually don't even need to do any math in order to realize that the resultant force from these two forces will point in the southeast direction. Consequently, the third force will need to point in the opposite direction in order to balance things out; the third force must be oriented in the northwest direction.

As can be seen in the diagram, the force pushing down the ramp is equal to

The force pushing the object into the ramp is

The force into the ramp will be equal to the normal force, thus

The net force pushing the woman down the ramp is

At the moment she starts slipping, the net force is equal to zero

Solving for :

Because the block is resting on an inclined surface, only a component of the gravitational force will contribute to the normal force between block and ramp. It can be assumed that the only relevant force acting on this block is due to gravity.

Since the normal force only acts in the direction perpendicular to the contact surface, the correct gravitational force component to take is the perpendicular component. This gives . By plugging in the known values for , and  , we get 

The tensions in both ropes are caused by the bucket being pulled down by gravity, hence the only relevant forces acting in this scenario are those in the y-direction. Since the bucket is being held up against gravity, the upward forces should balance exactly with the downward forces. The forces in the y-direction can be balanced like this:

, where  is the tension in rope 1. Thus we find that:

Conservation of momentum gives that the angular momentum of the golf swing is transferred into linear momentum for the ball, or that , so that the velocity of the ball immediately after the swing is .

Using kinematics, we know that , so that substituting gives .

Kinetic friction can be expressed as , so by substituting for  in the previous equation, we get

Plugging in known values and solving for  gives  and