As much as we all hate to admit, physics is incredibly useful. It describes the laws by which our universe operates. And depending on how you apply it, physics also has many ‘other’ uses. This article discusses elementary physics terms and kinematics, including basic two dimensional and projectile motion.
Vectors
Vector quantities have both a magnitude and direction. They’re usually given in the form of something along the lines of A = 5i + 10j. If you remember the Cartesian coordinate system, it’s easy to think of the i component as x and the j component as y. So, to represent our vector graphically, we would start from the origin and count 5 units to the right and then 10 units up.
As you can see, the magnitude of a vector would be the length of the straight line drawn from the origin to the resultant point. This forms a right triangle. Thus, by the Pythagorean Theorem, the magnitude is the result of taking the square root of the sum of each squared component. The magnitude of our vector A would be the square root of 5 squared plus 10 squared, or the square root of 125.
To add two or more vectors, add the corresponding components to get the resultant vector. For example, if we wanted to add our vector A and vector B = -9i + 1j, the result R would be R = (5 + -9)i + (10 + 1)j = -4i + 11j. You can multiply vectors in two different ways, but it isn’t necessary for this article.
To get the angle the vector makes with a coordinate axis, we must use the inverse tangent trig function. This is usually labeled as tan^-1 on most calculators. The tangent of an angle is defined as the y component divided by the x component, or the side opposite the angle divided by the non-hypotenuse adjacent side. So, the inverse tangent of the y component divided by the x component would yield the angle. In physics, the angle is usually expressed in degrees as opposed to radians.
Mass vs. Weight
One of the biggest things people confuse is mass and weight. Mass is defined as the actual amount of matter. The standard unit is the kilogram. It is a fundamental quantity; it doesn’t depend on anything else. An object has mass by virtue of its existence. Weight, on the other hand, depends on mass and the acceleration due to gravity. It is a unit of force, and from Newton’s first law, we know that force = mass x acceleration. The mass of an object will never change depending on its location. Weight changes slightly with different elevations on Earth and substantially on different planets, due to changes in gravitational acceleration.
Displacement vs. Distance
Displacement is really just an object’s position relative to a specified point. The standard unit is the meter. If you thought of yourself as the reference point and were looking at an object situated on a top shelf, the displacement would be the length of the straight line between you and the object. Distance has a slightly different meaning: how an object arrived at its current location. If you were to travel the full length of a circle with a 10 meter radius, using your starting position as a reference point, your distance traveled would be 20π meters (C = 2πr). However, your displacement would be zero since you returned to the starting point. Thus, displacement and distance are only the same when an object travels in a perfectly straight line.
Velocity vs. Speed
Velocity is the change in position (displacement) over time. It is usually given in meters per second. In calculus terms, it is the first derivative of the position function. Velocity is a vector quantity, so it has both a specified direction and magnitude. Speed is defined as the magnitude of a velocity vector. It’s a scalar quantity so no direction is needed. An object’s velocity could be 10 meters per second going north (V = 0i + 10j m/s), while its speed would just be 10 meters per second (Square root of 0 squared plus 10 squared is 10). Velocity is usually given in vector components.
Acceleration
Acceleration is defined as the change in velocity over time. Standard units are meters per second per second, or meters per second squared. Acceleration is the second derivative of the position function, or the first derivative of the velocity function. It is an indication how fast velocity is increasing or decreasing, and for our purposes, acceleration is always constant. When the acceleration and velocity have the same sign, it means an object is speeding up. When they have different signs, an object is slowing down. For example, if we were told that an object’s acceleration was -9 meters per second squared and was traveling in the positive direction, it would mean that for each second, we would subtract 9 meters per second from the initial velocity.
Basic Kinematic Equations
The following equations describe basic motion, assuming acceleration is constant and ignoring things like friction and air resistance. The object is treated as a particle; its shape, size, etc. are assumed to have no impact on its motion. In other words, using these equations alone makes a lot of unrealistic assumptions. Regardless, they are good approximations for the real world and have many applications.
Xi = initial position
Xf = final position
Vi = initial velocity
Vf = final velocity
t = time
a = acceleration
Vf = Vi + (a)(t)
Vf^2 = Vi^2 + 2(a)(Xf – Xi)
Xf – Xi = (Vi)(t) + (1/2)(a)(t^2)
Xf – Xi = (1/2)(Vf + Vi)(t)
Here’s a few examples of using these equations for one dimensional motion:
| An object has an initial velocity of 10 m/s with a constant acceleration of 5 m/s^2. What is its velocity after 3 seconds?
Vf = Vi + (a)(t) = 10 + (5)(3) = 25 m/s in the positive direction. |
| An object has an initial displacement of 10 m from the origin. Its initial velocity is -20 m/s with a constant acceleration of 2 m/s^2. What is its displacement after 15 seconds?
Xf = (Vi)(t) + (1/2)(a)(t^2) + Xi = (-20)(15) + (1/2)(2)(15^2) + 10 = -65 m from the origin. Since it started 10 m from the origin, the distance traveled would be 75 m. |
And so on. Picking the right equation to use depends on what you’re given. For more complex problems, you may need to solve two of these equations simultaneously.
Motion in Two Dimensions
These equations can easily be extended into multiple dimensions. For example, in two dimensions, we must simply think of an object as having an individual displacement, velocity, and acceleration along each axis. However, each component of acceleration must still remain constant.
For example, consider an object displaced 5 m from the origin at 53.13 degrees above the positive x axis. To get the individual components, we must recall some trig functions. The x or i component would be the magnitude multiplied by the cosine of the angle, and the y or j component would be the magnitude multiplied by the sine of the angle. The process is the same for velocities and accelerations given in these forms.
D = (5cos(53.13))i + (5sin(53.13))j m = 3i + 4j m
We could check this by taking the inverse tangent of (4j) / (3i), which would yield 53.13 degrees.
An example of a two dimensional motion problem:
| An object’s initial velocity is given by Vi = 3i – 4j m/s with a constant acceleration of A = 5i + 6j m/s^2. If it starts at the origin, what is its position after 10 seconds?
Here’s what we know: Position along the x axis: Position along the y axis: So, our displacement vector is D = 280i + 260j m. Taking the magnitude would give us 382.1 m, and the inverse tangent of (260j) / (280i) yields 42.88 degrees. Thus, our answer could also be expressed as 382.1 meters from the origin at 42.88 degrees above the positive x axis. |
Projectile Motion
I figured this would be especially relevant for Totse. As I mentioned, you’re ignoring air resistance and such, but this is still a good approximation.
In projectile motion, we imagine a particle traveling perfectly along an inverted parabola, with the x or i axis representing distance and y or j axis representing height. The x component of velocity is assumed to be constant, meaning the acceleration is zero. The acceleration along the y axis is the acceleration due to the force of gravity. This is approximately constant close to earth and is typically represented as -9.8 m/s^2. The negative sign indicates that the velocity eventually slows down to zero at some point, the arch of the parabola, and begins falling back down to earth with an increasing speed.
Typically you are given an initial velocity (magnitude) which is launched at a specific angle. Using our previous example, the initial (and final, since it’s constant) velocity along the x axis would be Vix = Vi(cos(θ)), and the initial velocity along the y axis would be Viy = Vi(sin(θ)). θ represents the angle of launch.
It may also be useful to know the range, or total horizontal distance, of an object. Likewise, it is also useful to know the maximum height a projectile reaches. For these two equations, make the acceleration due to gravity positive. The signs really don’t matter, they just indicate opposite directions.
Vi = magnitude of initial velocity
θ = launching angle of projectile
g = acceleration due to gravity
Range = (Vi^2)(sin(2θ)) / g
Maximum Height = (Vi^2)((sin(θ))^2) / 2g
An example of a projectile motion problem:
| A projectile is launched at an angle of 30 degrees above the ground with an initial velocity of 20 m/s. What is the range and maximum height of the projectile? How many seconds does it take for the object to hit the ground?
Here’s what we know: Range: Maximum Height: Total Travel Time: Xf = (Vix)(t) + (1/2)(ax)(t^2) + Xi We can check this equation by checking the final position of the y component at that time: If we had used more significant digits, we would get a number closer to zero. This makes sense in that the object has returned to the ground once it has traveled its maximum horizontal distance. |
Obviously these problems can be solved multiple ways. Happy launching Totse.
