Have you ever noticed that objects are in motion all around us? Everything from a tennis match to a bird flying in the sky involves motion. When you are resting, your heart moves blood through your veins. This gives rise to interesting questions: where will a football land if thrown at a certain angle? or how long will it take for a spaceship to reach outer space?
Kinematics is the study of the motion of points, objects, and groups of objects without considering the causes of its motion. To describe motion, kinematics studies the trajectories of points, lines, and other geometric objects, as well as their differential properties (such as velocity and acceleration). The study of kinematics is based on purely mathematical expressions that are used to calculate various aspects of motion such as velocity, acceleration, displacement, time, and trajectory.
In this lesson, we will investigate the words used to describe the motion of objects. We will study terms like scalars, vectors, distance, displacement, speed, velocity, and acceleration, that are often used to describe the motion of objects.
In order to describe the motion of an object, you must first describe its position - where it is at any particular time. You need to specify its position relative to a convenient reference frame. Earth is often used as a reference frame, and we often describe the position of objects related to its position to or from Earth. Mathematically, the position of an object is generally represented by the variable x.
Frames of Reference
There are two frames of reference:
a. Inertial frame of reference - This frame of reference remains at rest or moves with constant velocity with respect to other frames of reference. It has a constant velocity, that is, it is moving at a constant speed in a straight line, or it is standing still. Newton's laws of motion are valid in all inertial frames of reference. Here, a body does not change due to external forces. There are several ways to imagine this motion:
b. Non-inertial frame of reference - The frame of reference is said to be a non-inertial frame of reference when a body, not acted upon by an external force, is accelerated. In a non-inertial frame of reference. Newton's laws of motion are not valid. It does not have a constant velocity and is accelerating. There are several ways to imagine this motion:
Displacement is the change in the position of an object relative to its reference frame. For example, if a car moves from a house to a grocery store, its displacement is the relative distance of the grocery store to the reference frame which is the house in this case. The word "displacement" implies that an object has moved or has been displaced. Displacement can be represented mathematically as follows:
where Δx is displacement, xf is the final position, and xo is the initial position.
A vector is any quantity that has both magnitude and direction, whereas a scalar has only magnitude.
What is the difference between distance and displacement? Whereas displacement is defined by both direction and magnitude, distance is defined by magnitude alone. Thus, distance is a scalar quantity and displacement is a vector quantity.
Similarly, speed is a scalar quantity and velocity is a vector quantity.
The direction of a vector in one-dimensional motion is given simply by a plus (+) or minus (−) sign. Vectors are represented graphically by arrows. An arrow used to represent vector points in the same direction as the vector.
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In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations. It is any quantity that can be expressed by a single number and has a magnitude, but no direction. For example, a 20oC temperature, the 250 kilocalories of energy in a candy bar, a 90 km/h speed limit, a person's 1.8m height, and a distance of 2.0m are all scalar quantities.
Note, that a scalar can be negative, such as a -20oC temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows.
In order to describe the direction of a vector quantity, you must designate a coordinate system within the reference frame. For one-dimensional motion, this is a simple coordinate system consisting of a one-dimensional coordinate line. In general, when describing horizontal motion, motion to the right is usually considered positive, and motion to the left is considered negative. With vertical motion, motion up is usually positive and motion down is negative.
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Time
In physics, the definition of time is simple - time is change or the interval over which change occurs. The SI unit for time is the second, abbreviated 's'.
Let's understand how time relates to motion. We are usually interested in elapsed time for a particular motion, such as how long it takes a person to walk from his house to the park. To find the elapsed time, we note the time at the beginning and end of the motion and subtract the two. For example, the person may leave from his house at 11:00 AM and reach the park at 11:30 AM, so that the elapsed time would be 30 min. Elapsed time Δt is the difference between the ending time and beginning time,
Δt = tf - t0
where Δt is the change in time or elapsed time, tf is the time at the end of the motion, and t0 is the time at the beginning of the motion. For simplicity, we take the beginning time as zero i.e. motion starts at time equal to zero (tf = 0)
Velocity
Average velocity is displacement (change in position) divided by the time of travel,
\(v=\frac{\Delta x}{\Delta t}=\frac{x_f - x_o}{t_f - t_o} \)
where
v is average velocity; Δx is change in displacement; xf and xo are the final and beginning positions at times tf and to, respectively. If the starting time to is taken to be zero, then the average velocity is simply.
\(v=\frac{\Delta x}{t}\)
For example, if a person walks towards the rear end of a train. He takes 5 seconds to move -4m (the negative sign indicates that displacement is toward the back of the train). His average velocity would be
\(v=\frac{-4}{5} = - 0.8m/s\)
Instantaneous velocity is defined as the rate of change of position for a time interval which is very small (almost zero). If the object possesses uniform velocity then the instantaneous velocity may be the same as its standard velocity.
\(v_i = \lim \limits_{\Delta \to 0} \frac{ds}{dt}\)
where,
Speed
In everyday language, most people use the terms "speed" and "velocity" interchangeably. However, in physics, speed and velocity are distinct concepts. One major difference is that speed has no direction. Thus, speed is a scalar.
The average speed is the distance traveled divided by elapsed time.
Note that distance traveled can be greater than the magnitude of displacement. So, the average speed can be greater than average velocity, which is displacement divided by time. For example, if you drive to a store and return home in half an hour (30 minutes), and your car's odometer shows the total distance traveled was 6km, then your average speed as 12 km/h. However, your average velocity was zero, because your displacement for the round trip is zero. Thus, the average speed is not simply the magnitude of average velocity.
Instantaneous Speed is the magnitude of the Instantaneous velocity. It has the same value as that of instantaneous velocity but does not have any direction.
In physics, acceleration is the rate at which the velocity of a body changes with time. It is a vector quantity with both magnitude and direction. Acceleration is accompanied by a force, as described by Newton’s Second Law; the force, as a vector, is the product of the mass of the object being accelerated and the acceleration (vector), or F=ma. The SI unit of acceleration is the meter per second squared: m/s2
Acceleration is a vector that points in the same direction as the change in velocity, though it may not always be in the direction of motion. For example, when an object slows down or decelerating, its acceleration is in the opposite direction of its motion.
Acceleration is a vector in the same direction as the change in velocity, Δv. Since velocity is a vector, it can change either in magnitude or in direction. Acceleration is therefore a change in either speed or direction or both.
Mathematical representation of acceleration is:
\(a = \frac{\Delta v}{\Delta t}\)
where a is acceleration; Δv is the change in velocity; Δt is the change in time.
If a racehorse coming out of the gate accelerates from rest to a velocity of 15.0m/s due west in 1.80s, what is its average acceleration?
First, we draw a sketch and assign a coordinate system to the problem. This is a simple problem, but it always helps to visualize it. Notice that we assign east as positive and west as negative. Thus, in this case, we have a negative velocity.
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We can solve this problem by identifying Δv and Δt from the given information and then calculating the average acceleration directly from the equation:
\(a = \frac{\Delta v}{\Delta t}\)
⇒ \(a = \frac{-15 m/s}{1.50 s}\)= - 8.33 m/s2
Motion with constant acceleration
Constant acceleration occurs when an object’s velocity changes by an equal amount in an equal time period.
An object with a constant acceleration should not be confused with an object with a constant velocity. If an object is changing its velocity -whether by a constant amount or a varying amount - then it is an accelerating object. And an object with a constant velocity is not accelerating.
There are four kinematic equations that describe the motion of objects without consideration of its causes. The four kinematic equations involve five kinematic variables: d, v, v0, a, and t.
where,
d stands for the displacement of the object;
v stands for the final velocity of the object;
v0 stands for the initial velocity of the object;
a stands for the acceleration of the object;
t stands for time for which the object moved.
Each of these equations contains only four of the five variables and has a different one missing. This tells us that we need the values of three variables to obtain the value of the fourth and we need to choose the equation that contains the three known variables and one unknown variable for each specific situation.
Equation 1
v = v0 + at
Equation 2
d = \(\frac{1}{2}\)(v0 + v)t or alternatively, vaverage = \(\frac{d}{t}\)
Equation 3
d = v0t + (\(\frac{at^{2}}{2}\))
Equation 4
v2 = v02 + 2ad
