Linear Motion
The word 'linear' has an important meaning in math or physics. It means 'a one-to-one relationship'. One-to-one means that the scaling remains the same no matter how small or big either of the relating values get. In mechanics, 'Linear Motion' means 'motion in a straight line'. In reality it is hard to do. This is due to bumps and bends. Motion is usually caused by some kind of motivation, like a thrust or push or pull. In this here write-up we will ignore the motivation whatever it is. This means we are talking about Kinematics. If we were to look at this reasons why the motions came about we would be talking kinetics.
The Demanded Changes Through Time. Distance and Speed
Time
Time is an accumulation of a registered event during which other events could be registered. These events could be anything, like explosions, experiments a race, and other stuff that could be observed or happen. Change occurs through time constantly because thing don't stay still in many places at once.
In physics things are rated according to how much time it takes, and sometimes or a lot some things will happen again at almost equal times. One example is a day. A day takes equal times to go through: 24 hours. I'm sure you can get the rest. You usually measure time with a clock or watch. The units of time are usually seconds, minutes and hours
You may need tutorials on solving algebra: Simple Algebra
and vectors: Vectors
Distance
A distance is total length of the space that is traversed or occupied. This traversing could be could happen by necessity in an amount of time and can follow any shape like a circle, an arc, a square path etc. You measure distance with a ruler or tape rule among other things you could use. The known units of distance are usually meters, centimeters, kilometers, miles, feet, inch, millimeters etc
Speed
Speed is the amount of distance traveled in a chosen unit of time. It is used when an object is in motion
Some thing move just as fast as you can see them, but some things are too fast to see. To calculate a speed you take it's position before as d1
and the position after as d2 then speed s is
s=(d2-d1)/t
The units of measurements for speed are m/s, km/h, mile/h, feet/s and others. Like time some things
in nature have constant or chosen constants for their speed. One example is light which always travels at 3x108 in a vacuum.
Vectors Again: Change Scale And Direction
It turns out elements of motion can be spelt out in vectors, because you can see they have a direction. Like to move an object has to choose a direction.
S we have a 'velocity' and a 'displacement'. Displacement is a distance in a direction meaning a straight path, and velocity is the speed of an object
in a certain direction Look at the diagram below:
In the first one, a thing moves from A to B to C. The paths AB and BC are straight paths, so does that mean the displacement is AB plus BC, whatever AB and BC are?
Well, not so? The displacement is actually AC. You have to use solutions in triangle to find it. Look at the diagram below:
s is the starting point and e is the endpoint. The displacement is the distance and direction of s towards e. To find the total distance, you have
to add the length of all the individual paths from s to e. Now look at this one:
This one shows an object moving from A to B at a constant speed. This means its velocity is constant, Nah! It is not. It is actually changing consistently.
It is changing constantly because the path is not a straight path. So even though this vehicle manages to maintain an equal speed all the time between A and B,
it has failed to maintain the same velocity. Lets assume v1 and v2 are close together. The change in velocity between
them is below:
The difference is vector subtraction as opposed to algebraic subtraction. You use a solutions in triangle to do it. As you can see, we did not simple just minus the speeds as the answer will be zero and the two arrow points will be touching each other.
The manipulation above is different from doing a resolution where we find the resultant between two vectors. But the mathematics is the same.
Circuits Or Partial Circuits
Recall just now I said that a path could change the velocity because of a change in direction. Now we will consider motion on
a curve with constant speed. We will do a short introduction to acceleration in the next part, but for now consider the diagram below:
Assuming this change happens during a short time such that the angle there is very nearly zero or could be taken as that. Now each part of a curve
has a tangent and this tangent could be attached to a particular circle, or you could build out of that point or infinitesimal
portion a circle that can have or own the same tangent. Look below:
The 'owning circle' has been generated in this diagram showing its tangent and also radius of curvature. This is a line showing
how big or small the owning circle is. This line in infinite if the path of motion is a straight line. Now as before imaging that v1 and v2 are close together
which makes the two radii super-imposed on each other making them look like one line. When you look again at the change in velocity using the above diagram:
You will see under close observation that the two velocities must align with the tangents at their the points where the occur. Now since the are very close points it means
they essentially align or super-impose one another. The change in velocity therefore falls into the radius of curvature of the circle. You see, since the velocities
have equal magnitudes(speeds) the triangle they form is an isosceles where the top angle is zero making the vector difference of v1 and v2
perpendicular to v1 and v2.
This is the case for all circular motion where the speed is constant. There is a change in velocity towards the center of motion.
Graph-itees
To find the distance moved when you have the velocity-changes, you can use a graph to do it. You see, the area covered by a graph tell a lot. In this case
a Velocity-Time graph can tell you the total displacement. Look below:
You can see that the velocity starts from zero at a time set to zero and starts to increase at a consistent rate, then it reaches a point where it sharply starts to drop speed at a consistent rate. It then comes down to a zero velocity and still continues to drop speed. How can it do that?
It means that it has changed it direction and start moving opposite its original direction of motion.
So this in turn means that it start to uncover some of its originally covered ground. Just say you walk 40 meters forward and now you walk 15 meters backwards making you displacement along that line 25 meters forwards.
Going back to the diagram, the velocity again changes its rate such that its velocity to increase in the former direction. Imagine a back doing a reverse in
his car very fast the starts to reduce the speed of reversing. You can rightly claim that the speed of going forward driving is increasing until the reversing stops, which is the case
again because the velocity come again to zero in magnitude. Now there is a point above the zero line where there starts a line parallel to the t-axis
where the velocity remains constant in value. Here is a time during which, for instance, the speedometer registers a consistent reading in speed. After that time the
velocity finally drops to a stop, which might mean that the person has reach where he or was looking for. The total displacement can be found by finding all the areas below and above the
t-axis and taking those above as positive and those below as negative and doing a sum. For instance look below:
The area of the first triangle is:
+1/2x10x3=+15
The area of the second one is:
-1/2x5x(4-3)=-2.5
The area of the third is:
+1/2x15x(10-4)=+45
The total traversed displacement is:
+15+(-2.5)+(+45)
You can do the sum.
Equations of Linear Motion
The equations of linear motion when acceleration is constant or uniform are below:
v = u + at
v2=u2+2as
s = ut + (1/2)at2
u is initial velocity, v is final velocity,
s is the displacement,
a is the uniform acceleration,
t is the time taken
For other studies in Physics go to: Physics and the needed math: Math
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