In this episode, a train crash occurs causing fatalities and a leak of toxic chemicals. It's up to the FBI team to help rescue the living victims as well as figure out the cause of the crash. Math and physics play a role in both of these tasks.

In order to help rescue the living victims in one of the rail cars, Charlie uses small robotic cars known as *swarm bots*
to survey the situation and map out a route for the rescue team. To determine the cause of the crash, Charlie utilizes some basic physics to deduce
that the conductor was lied to about the mass of the train and cargo.

In this website, we'll focus on some of the math and physics that Charlie needs in order to figure out the cause of the crash. However, if you wish to learn about swarm robotics, the following are a few places to begin.

1. http://www.swarm-robotics.org/2. http://www.swarm-bots.org/

3. http://en.wikipedia.org/wiki/Swarm_robotics

Suppose an object is moving along a straight line. We can fix a reference point and graph the position of the object (relative to this reference
point) over time. Let s(t) denote the function which has this graph. The value of s at a given time t* is the *displacement* of the object at time t*. For example,
if s(t*)=5 then the object is 5 units in front of the reference point at time t*. If s(t*)= -5, then, at time t*, the object is 5 units behind the reference point. Therefore,
if t_{1} and t_{2} (with t_{1} < t_{2}) are two times, we have that the *average velocity* of the object between time t_{1} and t_{2}
is

If Δt=t_{2}-t_{1}, we can rewrite the above equation as

For those of you that are familiar with basic calculus, notice that the formula for average velocity would look like the definition of a derivative if there were a limit thrown in.
In fact, if we take the limit as Δt goes to 0, we obtain *instantaneous velocity* at time t_{1}. That is,

We have that s(t) is displacement, s'(t)=v(t) is velocity and v'(t)=a(t) is acceleration. But what happens if we take another derivative? What physical quantity does a'(t)
represent? The answer is *jerk*. You can definitely feel this when riding in a car that quickly changes how much it's accelerating!

As indicated above, the velocity function is denoted by v(t).
Analogously, we can define *average acceleration* (over time Δt) and *instantaneous acceleration* (at time t) by

respectively.

Let's consider the example s(t) = -(t-3)^{2} + 9. This describes the motion of an object which is at the reference point at t=0, then moves forward, changes direction and
eventually ends up behind the reference point. We also have that v(t)=-2(t-3) and that a(t) = -2. That is, the object with position described by the function s(t) has constant
acceleration. Below are the graphs of s(t), v(t) and a(t).

Suppose that an object X is experiencing linear motion. The position of X over time is described by the function s(t) = t

- Sketch the graph of s. Describe the motion of X.
- What are v(t)=s'(t) and a(t)=v'(t)? Sketch the graphs of v and a. Now describe the motion of X in more detail.

When acceleration is constant, one can derive equations of (linear) motion by solving the differential equations described above. For example, consider the equation
a(t)=v'(t). If acceleration is a constant a then we have the equation a = v'(t). Separating variables and integrating from t_{i} to t yields the equation a⋅(t-t_{i}) = v(t) - v(t_{i}).
Thus, if we put Δt = t-t_{i} and v_{i}=v(t_{i}), we have that v(t) = v_{i}+aΔt. Using similar techniques, we can get the following equations of linear motion:

In the above equations, v

One final remark is needed: Be sure to be consistant with units when using the equations of motion! For example, don't measure acceleration in m/s^{2} and velocity in km/h. Either
measure both quantities using meters and seconds or both using kilometers and hours.

Force is a vector quantity. That is, like velocity and acceleration above, it has both a magnitude and direction. When a force is applied to an object, it causes the object
to accelerate in the direction of the applied force. The relationship between force and acceleration can be made precise with the famous equation

Here F is force, m is mass and a is acceleration.

Of course, many forces can act on the same object. It is even possible that the net force (i.e. the vector sum of all of the forces acting on the object) is 0. In this case, the object will stay put. When multiple forces act on an object we have the equation

A *free body diagram* is helpful in keeping track of all the forces that act on an object. It is a diagram where the object is represented by a box (or other simple
shape) and includes arrows of different lengths and directions to represent the various forces that act on the object. Below is an example of a free body diagram for an object
being pushed along a table by an applied force F_a. F_g, F_N and F_f denote force of gravity, normal force and force of friction respectively.

The SI unit for force is the newton (N). One newton is equal to one kg⋅m/s^{2}. It is again very important to be careful with units when using the force
equations above.

A 2kg block is placed on a 30

- Draw a free body diagram and label all forces.
- Calculate the normal force.
- Will the block stay put or will it begin to slide down the surface? Explain.

- Let μ be the coefficient of static friction between a block and an inclined plane. Determine an expression (in terms of μ) for the maximum angle of incline
of the plane before the block starts to slide.

Imagine that you're the conductor of a train. Your train weighs 600 000 kg and is moving at a speed of 120 km/h. Suddenly, you notice a stalled truck on the tracks 150 m away. You have to hit the brakes. Your train slows down due to kinetic friction, but will it stop in time?

- Draw a free body diagram of the situation and label all forces.
- Assume that the coefficient of kinetic friction between the train and tracks is 0.45. Does the train stop in time?
- Now suppose that you can control exactly how much force is applied to the train. What force must be appied to the train to cause it to stop just in the nick of time?

Here are some references and places to learn more:

1. Giancoli, Douglas C.2. http://en.wikipedia.org/wiki/Kinematics

3. http://en.wikipedia.org/wiki/Force

4. http://scienceworld.wolfram.com/physics/Friction.html