The net force experienced by m1 can be characterized as:

 where T is the tension force along the string and mg is gravity.

because  is also equivalent to  , we can express this equation as

 .

The net force experienced by m2 can similarly be expressed as:

 and  so 

From this point the problem is a system of equations.

If we solve for T in the first equation, we end up with .

Then we can plug into the second equation like so:

then:

finally we can isolate a:

when we plug in the masses given by the problem, we get:

Let's interpret each graph.  

(A) is not correct because it shows Beverly riding her bike, being in the store for a period of time, and riding back at the same speed she was going when she rode to the store. Then she goes backward in time (not possible) and ends up at the same time and location she started at.  

(B) She rides her bike to the store, stays there for a length of time and rides back.  At  (arbitrary place) she has a flat time and walks home. Notice that the slopes of the position vs. time sections before and after the store are similar. The only difference is that one is positive and one is negative, indicating a positive and negative velocity respectively. Remember that a change in position over a change in time is velocity. Now she walks home which means she is traveling slower than on her bike. This means that the slope of  vs.  is smaller. It will take her a longer time to walk back so this portion has a smaller slope. She ends up at the same  position she started from but at a later time.  

(C) This plot is correct if we were looking at her velocity vs. time. She starts out with a higher speed (positive) and then it's close to zero at the store. She then has a negative velocity that has the same magnitude as her trip to the store until she gets a flat and then her velocity is smaller but still negative as she walks home.

(D) This graph would be correct if she did not have a flat and rode home the entire way.

The sum of two vectors written in component form will result in a new vector whose components are the sum of the components of the added vectors, written as:

The magnitude of a vector is given as the square root of the sum of the squares. Written completely as:

For our example, the solution is written in full as:

By definition, the dot product of two vectors can be related to their magnitudes and the angle between them as follows:

Given the angle between the two vectors is , we can calculate the dot product to be written explicitly as:

Note that the unts are  since the dot product involves multiplying two meters together.

Displacement is not the same as distance. Over the course of the problem, the ball starts and ends in the same place. While the ball covered a distance of 14 meters during the problem, it's displacement is 0 meters.

If we say going to the right is positive and to the left is negative we have:

We calculate displacement this way because it is a vector, meaning it has both magnitude (how far it moved) and direction (which direction did it move). Distance refers only to magnitude.

Velocity and acceleration are both vectors which have a magnitude and a direction. We use the sign of a value to notate direction and in this situation we will say north is positive and south is negative. We use the equation 

Where  is the final velocity,  is the initial velocity , the acceleration is , and the time of the acceleration is .

Plug in known values and solve.

The sign of our answer is positive, meaning the car is still traveling north after the acceleration.

Displacement is the difference from the person's initial position and final position. In this problem, since the man is walking on orthogonal axes (x and y), we determine his displacement by using the pythagorean theorem. 

Distance travelled is the sum of all motion, which is:

The difference between distance and displacement is:

Measuring the total displacement means finding the norm of the vector that starts from the original spot to the final.

Since the object moved 2 feet east and 1 feet north , the norm of this vector can be found by doing:

For this question, we need to find out which graph represents constant velocity.

First, let's recall that velocity is defined as the amount of displacement that an object undergoes in a given amount of time. Hence, a graph in which the displacement is changing at a constant rate is showing a velocity that is constant. If this line is not straight in a graph of displacement vs. time, then the velocity is changing during the object's travel.

In a graph of acceleration vs. time, a straight flat line indicates that acceleration is not changing. And since acceleration is defined as a change in velocity with respect to time, a flat line indicates that the velocity is changing at a constant rate. Thus, in this scenario, velocity would not be constant (since it is changing at a constant rate). In an acceleration vs. time graph that contains an upward facing straight line, this indicates that the acceleration is increasing at a constant rate. And if acceleration is increasing at a constant rate, then the velocity would be increasing at an even greater rate (an exponential rate).

We will use Newton's 2nd law to solve this problem:

There are two forces acting on the sledder in the direction of their motion: friction and gravity. We are given the frictional force, so we just need to calculate the component of gravity in the direction of the sledder. We will start with the force of gravity:

And then to get the component in the direction of the sledder's motion, we will use the sin function:

If you're wondering why we use sine, think about the situation practically. As the angle grows, the hill gets steeper, and their will be more gravitational force in the direction of motion. Hence, we use sine. Now adding these forces together to get the net force:

We subtracted the friction force since it is always in the opposite direction of motion.

Now solving for acceleration: