This is a position versus time graph.  To find the velocity of an object at a particular time, determine the slope of the graph at that time.  Using the points  and  we can determine the slope of the graph at 25 seconds since it is a constant slope.

Velocity is equal to a change in displacement per unit time. Consider a basic velocity vs. time graph that depicts a constant velocity, given by a straight horizontal line. How can we determine the displacement during a given time interval? Use the kinematics equation:

Acceleration will be zero, allowing us to reduce the equation. We can also assume that there is zero initial displacement.

So, the displacement will be equal to the velocity multiplied by the time. In the graph, velocity will be represented by the vertical location and time will be represented by the horizontal location of any point. Displacement will be equal to the height of the graph times the width of the graph, or the area of the rectangle created by the constant velocity over a unit of time.

Though this is just one example, displacement will always be equal to the area under the curve of a velocity vs. time graph.

In calculus terms, velocity is the derivative of the function for displacement in terms of time. To find the displacement, one must take the integral of the velocity function. The result is the area under the curve of the velocity function.

This is a position versus time graph.  The slope of the line of the graph tells us the velocity of the object. 

For the first  the object has a constant positive slope indicating that the object is traveling at a constant positive speed.

From  to  the object remains at the same position which means that the object has stopped. 

From  to  the object has a constant negative slope meaning that the object is traveling at a constant negative speed.  

At , the object switches directions and now has a constant positive slope meaning that the object is traveling at a constant positive speed.

Begin by identifying what forces are acting on the car.  To start, the car has weight (the gravitational force) acting on it.  This force points down.  The car also has a normal force acting on it keeping it from falling through the road.  This force points upward and is equal in magnitude (size) to the weight force.  The car is accelerating, in the horizontal direction, which means that the forces must be unbalanced.  If the forces were balanced in the horizontal direction, the car would either not be moving or moving at a constant velocity.  The car likely is also undergoing friction which would be less than the accelerating force.  Therefore diagram C is correct as there is an unbalanced force in the horizontal direction with a friction force opposing the motion.

Begin by identifying the forces on the book.  The book has a weight (gravitational force) pushing down on it.  The book is sitting on a desk, therefore the desk has an upward normal force acting on the book keeping it from falling through the desk which is equal in magnitude (size) to the gravitational force.  The book is being pushed from the side meaning there are horizontal forces at work.  The first is an applied force, the second is a force of friction against the motion.  Since the book is moving at a constant velocity, the book is not accelerating.  This means that the forces must be balanced.  Therefore Diagram B is the correct answer as the horizontal forces are balanced.

Once the ball leaves the bat, there is no longer a contact force applied on the ball in the upward direction.  There are no horizontal forces at work either.  The only force acting on the ball is gravity.  Gravity points in the downward direction. 

If the drop is at rest, then that means that the net forces acting upon it are equal to zero. There are two forces acting on the drop: the force due to gravity and the frictional force of the paint on the wall.

Mathematically, we can set up an equation for the net force:

The forces of friction and gravity are going to be equal and opposite, causing the drop to remain still on the wall.

In this diagram,  represents the force due to friction. The equation for the force due to friction is , where  is the coefficient of friction.

In this case,  represents the normal force. We can re-write the equation for friction:

 can be re-written in terms of the angle, but will always need to be multiplied by the coefficient of friction in order to give an equation for .

The other equations are true.

   and  form a right angle, so the Pythagorean theorem applies.

 is the normal force, which is, by definition, equal and opposite the vertical force of gravity.

 is the total force of gravity, which will be equal to the mass times the acceleration of gravity.

 - the triangle formed by , , and  is similar to the triangle formed by the surface, , and , meaning that these angles must be equal.

For this problem, we need to break the forces into horizontal and vertical components. The only difference with an inclined plane is that you have to translate the horizontal plane to be parallel to the surface upon which the object is traveling, and the vertical plane to be perpendicular to the surface upon which the object is traveling.

Luckily, this is already done in the diagram, with the force of gravity () broken into the vertical component () and horizontal component ().

Now, we need to sum the forces in each plane. The forces in the vertical plane will be perpendicular to the surface:  and .

Since  and  are opposite and equal forces:

There is no net force in the vertical plane. This makes sense, as we would not expect the object to have any movement perpendicular to the surface on which it is sliding.

Now we can sum the horizontal forces:  and .

Note that  is positive and V is negative. This is because  represents the force of friction, and is opposite to the force of .

Since there is no force in the vertical plane, this gives our final net force: 

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  in the diagram.

Observe that  and  are equal, but opposite forces.

If we can solve for , then we can find .

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

If we plug in a 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 , we can return to our original equation to solve for .

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

Substitute this into our equation for .

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