Newton's second law states that .

If Derek is pushing with  of force, then we should be able to solve for the acceleration of the crate.

Derek observes that the crate is acceleration at a rate of , rather than the expected . An outside force is acting upon it to slow the acceleration.

The equation for the net force on the object is: . We also know, from Newton's second law, that , where the resultant force and acceleration are the values actually observed.

Plug in the information we've been given so far to find the force of friction.

Subtract  from both sides to find the force of friction.

Friction will be negative because it acts in the direction opposite to the force of Derek.

Newton's third law states that when one object exerts a force on a second object, the second object exerts a force equal in size, but opposite in direction to the first. That means that the force of the hammer on the nail and the nail on the hammer will be equal in size, but opposite in direction.

Since the hammer exerts  of force on the nail, the nail must exert of force on the hammer.

For the net force, we add up all the forces: .

Since force is a vector, the direction of the action matters. We will make leftward motion negative and rightward motion positive. Michael is pushing with  to the left, making his force equal to . Annie was pushing with  to the right, so her force will remain .

We can find the net force by adding the individual force together.

If the object has a constant velocity, that means that the net acceleration must be zero.

In conjunction with Newton's second law, we can see that the net force is also zero. If there is no net acceleration, then there is no net force.

Since Franklin is lifting the weight vertically, that means there will be two force acting upon the weight: his lifting force and gravity. The net force will be equal to the sum of the forces acting on the weight.

Since we just proved that the net force will equal zero, we can say .

We know the mass of the weight and we know the acceleration, so we can solve for the lifting force.

The formula for force is .

We are given the mass, but we will need to calculate the acceleration to use in the formula.

We know the initial velocity (zero because the box starts from rest), final velocity, and distance traveled. Using these values, we can find the acceleration using the formula .

Plug in our given values and solve for acceleration.

Divide both sides by .

Now we know both the acceleration and the mass, allowing us to solve for the force.

In this problem there will be two forces acting upon the airplane: the weight of the plane (force of gravity) and the lifting force. Since we are looking for the minimum force to lift the plane, we can set the two forces equal to each other: .

We can calculate the gravitational force using the mass.

Returning to the original equation, we see that the lifting force must be .

The formula for force is . Plug in our given values and solve:

We need to look at the net force, which is the sum of the other forces:

Since Alexis and Miguel are pushing with the same value but in opposite directions, make one of these forces negative:

Therefore, the net force upon the object is .

Use Newton's third law to solve this question. The force of the dog on the chew toy is equal in magnitude, but opposite of direction, to the force of the chew toy acting on the dog.

That means that .

Using our given values for find the force of the chew toy.

The normal force is always perpendicular to the surface upon which the object is moving, and is pointed away from said surface. That means we are looking for the value for Z in the diagram.

Observe that Z and Y are equal, but opposite forces.

If we can solve for Y, then we can find Z.

We can use our understanding of trigonometry to find an equation for Y.

If we plug in  for the angle, we see:

Since we are solving for Y, we can multiply both sides by W.

Now that we know an equation for Y, we can return to our original equation to solve for Z.

From here, we can use Newton's second law to find the value of W, the total force of gravity.

Substitute this into our equation for Z.

Now we can solve for Z using the values given in the question for the angle, mass, and gravity.