Linear Motion Equations | Simple example & Two Stage Rocket Example

Linear Motion Equations | Kinematics | Linear Motion Formulas

What: “Kinematics” deals with objects in motion. Linear motion means that the objects move in a straight line, which simplifies the mathematics.

Why: If you know some information about the movement of an object you can figure out other details.

How: You use two basic linear motion equations.  They include terms where you can use calculus for more complicated situations (velocity and acceleration)

1.Two fundamental equations
v = v0 + at
x = x0 + v0t + (1/2)at2

These are two very useful equations that you can use for early linear motion problems.

Why: If you understand these equations and have them memorized, you can solve many problems without memorizing a lot more equations. In general, I prefer to memorize less if possible.

How: Certain details of a situation, such as starting at “zero” or having a zero acceleration will simplify these equations. And you can also combine these two equations to do more.

Wait….. I thought I needed 10 equations, not 2. If you understand all the parts of these two equations and can combine them, you will generally not need more equations for many problems. If you

2. Parts of the equation

v: velocity (later on, after v0)
v0: starting velocity, “v naught”
a: acceleration
t: time

x: position
x0: starting position

How: you see some of the variables show up more than once in these equations. In the second equation, you do see the time is squared in the final term. Sometimes gravity is the acceleration that you will use.

Note: you will want to use SI units,
meters and seconds
If you do that, the outputs will also be in SI units

3. Examples

Simple example:

A person runs at a velocity of 5 m/s, how long will it take to run 20 m?

x = x0 + v0t + (1/2)at2

You can take the starting position to be 0 and there is no acceleration mentioned, so you take that to also be 0.

The equation simplifies to

x = v0t

“How long” asks for the time, so you want to solve the equation for time

x = v0t
Divide both sides by v0

x/v0 = t

20 m/(5 m/s) = 4 s

Harder example:

A rocket ascends vertically (from rest) with an acceleration of 4.00 m/s^2 until it runs out of fuel at an altitude of 1000 m (1.00 x 10^3 m). At that point, its new acceleration is due to gravity, downward.

a) What is the velocity of the rocket when it runs out of fuel?

v = v0 + at
x = x0 + v0t + (1/2)at2

For many problems that are a bit more complicated, you will use the two kinematic equations.

Let’s start with the second equation

1000 m = 0 + 0 + (1/2)at2

1000 m = (1/2)at2

2000 m = (4 m/s^2)t2
5000 s^2 = t^2
t = 5000^(1/2) s
~70.7 s
Note! Keep this stored in memory, you don’t want to round until the end.

We can put that result into the first equation to get the velocity at this point
v = 4 m/s^2 * 5000^(1/2) s
v ~ 283 m/s

b) What is the maximum altitude of the rocket?

Probably need to use both equations, definitely need the equation with distance

To keep track carefully, it can be useful to use subscripts for variables, especially since there are two stages of motion here (with fuel and without)

red and blue!
x = x0 + v0t + (1/2)at2
x = 1000 m + v0t – (1/2) 9.8 t2

v = v0 + at
0 m/s = 283 m/s – 9.8t
-283 m/s = – 9.8t
(-283 m/s)/(-9.8 m/s^2) = t
t ~ 28.9 s

x = x0 + v0t + (1/2)at2
= 1000 m + v0
1000 m + 283(28.9) + 0.5 * -9.8 * 28.9^2


~5080 m (3 significant figures)

Let me know what questions you have, I could make a followup video

Tutoring Physics, Revisiting Linear Motion Equations

Aiden’s class is getting into conservation of energy. So gravitational potential energy and linear potential energy. Possibly rotational motion which would involve angular velocity, angular acceleration, and moment of inertia (not sure when/if they will cover that) though.

Momentum also came up again in one problem.

Most important thing we went over was the return of two equations from kinematics early in the class that are being used again.

Linear Motion Equations


The less wise of a group of monks, after hearing the equation x-x0=v0xt and rushing away from the physics lecture, thought that if they gave building blocks a simple push they would continue moving until stopped by another push.

They believed that by utilizing this piece of knowledge they could easily build a grand monastery of marble blocks.

They gathered up a group of rather gullible monks, who had not taken much physics, and went out to the quarry.

Unfortunately when they gave a block of marble a push, it did not start moving. Bewildered, the monks did not understand what was happening.

Fortunately, a bishop/physicist came along and explained that horizontal movement actually is also affected by vertical components through friction.

He stated that when something is in contact with a surface, the normal force and friction coefficient comprise a force opposite to the direction that the object is being moved in, with the formula f=μN.

This wise man said that the normal force also is changed if there is a slope of an angle theta. One of the less knowledgeable monks asked about force triangles, but the wiser bishop/physicist said that they are unnecessary if the physics is truly understood.

Rocket Explosions, Probability, and Compound Interest

If there are 1000 parts for a space shuttle and each of them has a 99.9% chance of working, then the chances for no malfunction at all is only about 37%.

By multiplying .999 by itself 1000 times without rounding, the true result is found, but if each time it had been rounded up a crew might have piloted the spacecraft and met their doom in an explosion.

When manipulating equations, the numbers should also not be plugged in until the end.

Heat Flow

Heat naturally flows from warmer to cooler.  Things tend towards equilibrium.

Therefore heat vents are most efficiently placed on the floors of rooms so the heat might rise up throughout the room.

If the vents were on the ceiling, then all the heat would stay towards the top of the room.

And fans are then placed on the ceiling to circulate the air around more efficiently.

All this works according to the formula H=K*L/A(TH-TC) where K is the coefficient of thermal cooling for a substance.

Physics of White Water Rafting Bernoulli Pressure

If you go ever go white water rafting you’ll start out in a nice calm area, kind of like the happy singing place on Splash Mountain in Disneyland.

The river is wide at this point and the velocity of the water is fairly low.

Further down the river though the river will narrow and rocks will be there.

According to Bernoulli, p1 + (1/2)ρv12 = p2 + (1/2)ρv22


p is pressure
ρ (rho) is density

and the narrowness will therefore increase the velocity to allow the same amount of water to flow in the same time it takes at the calmer points in the river.

More water has to go through a smaller area so it will need to go faster.

Tension Free Body Diagram

If a vine is hung over a tree branch in the forest and two monkeys are hanging on both ends then the tension in the vine will be equal throughout the whole rope, since if it wasn’t the vine would snap or go slack.

The tension force for one monkey’s side will be equal and opposite to the tension force on the other monkey’s side.

The vine itself will also require a different amount of force to slide across the top of the branch depending on how the vine and the tree branch interact.

The force required for movement is unique depending on the materials (which defines a coefficient of friction).

Objects will have different coefficients of friction, denoted as mu, based on how they act together.

In analyzing situations it is good to draw a picture or “free body diagram” and decide what the axes are, including which directions should be understood as positive and negative.

The free body diagram should  only be applied to simple situations (generally one or a few objects involved).

When the situation is complicated, like the effect of huge numbers of electrons hitting an object, other theories come into play.

Even with fifty forces that are three dimensional, you would have to add up 150 components (x, y, z for each) and that wouldn’t be fun.

Changing Coordinate Systems to Make Problems Easier

One day a physicist comes across a perplexed student in the library.

It seems that the student had only learned to place his axes with the positive y axis going up and the positive x axis going to the right.

Unfortunately this system wasn’t working so well with a situation involving pulleys and curved directions.

The physicist decides to help the engineer by telling him that the x-y coordinate system can be changed to better suit the problem at hand, much like polar coordinates can be more convenient than cartesian coordinates.

If an object travels in a curved path he reasons that the x-y coordinate system may also be curved in parts for simplicity.

Gravity may be thought of as positive or negative and up may be considered down.

The physicist also says that simply using common sense to check an answer will often times be effective, for instance if a block is sitting on a wall with no forces it will probably just stay there.

Newtons Second Law F=ma

As the village blacksmith walks along one day he sees a group of monks circling around an apple tree and chanting something about the truth being expanded.

When he stops and listens he learns that a man named Newton became inspired by falling of apples.

The monks explain that Newton then came up with the equation ΣF = ma (where F and a are both vectors) which means that the sum of the forces in a direction is equal to the mass, not to be confused with weight, multiplied by the acceleration, one example being gravity, in a direction.

The F and a are vectors, indicated by bold.

The monks are rejoicing since they had only previously been able to understand the motion of objects.

Now they can understand interaction of objects with mass through the use of vector components, the old testament of the truth, and their newfound equation.

Circular Motion Centripetal Acceleration Arc Length

One day a little kid is playing with a yo-yo and swinging it around in a circle for his own amusement.

Suddenly a crazed physicist walks up and decides to explain the physics of the circular motion.

First he asks the kid to spin the yo-yo around without any acceleration and the kids tries, but this turns out to be a trick question since there always is an acceleration towards the center (centripetal or center seeking force) because the velocity changes directions.

The physicist next shows how components can be used for measuring velocity vectors.

He then explains that describing certain aspects of circular motion with rectangular coordinates is pure silliness and that sometimes polar coordinates with an angle and a radius are much simpler.

He notes however that unlike in navigation, the angle is measured from the positive x-axis in physic.

Finally the physicist gives a nifty trick (s=rφ) for finding the arc length of pizza, or any circle, if one knows the radius and angle phi, but speaking of pizza he thinks about how good some would be and walks off.