Position-time graph, speed and velocity

 Kinematics is a field of mechanics concerned with the study of motion. Any kinematics study aims to create complex mental models that may be used to describe (and eventually explain) the motion of real-world objects.

1. Position time relation

A moving object’s displacement is proportional to its velocity and time. Quicken your pace. Go a step beyond. Move for a longer duration. Go a step beyond. Because velocity is now exactly related to time, acceleration complicates this straightforward scenario. It sounds ridiculous when you say it out loud. “Displacement is directly proportional to time, as is velocity, which is also directly proportional to time.” Because time is multiplied twice, displacement is proportional to the square of time.

• When acceleration is constant and the initial velocity is zero, displacement is proportional to the square of time. Any initial velocity, as well as how the velocity was changing, would have to be considered in a truly general statement. As a result, the proportionality statement is extremely clumsy. When the acceleration is constant, the displacement is proportional to the square of time and is directly proportional to time. A quadratic function is one that is both linear and square, allowing us to compress the previous statement significantly. When acceleration is constant, displacement is a quadratic function of time.
• Proportionality statements are helpful, but they aren’t as versatile as equations. For this situation, we still don’t know what the proportionality constants are. Algebra is one method for determining them.

                Average velocity     v = ∆s /∆t

Here ∆s is the displacement

∆t is small time duration in which displacement occurs

               Expand  ∆s to s -s0 and condense ∆t to t = 

 v = (s − s0 )/t

              Solve for the position . s = s0 + vt       ……..   [a]

The average value of a quantity is halfway between its final and initial values when the rate of change is constant.

vav = ½(v + v0)

Distance travelled by average velocity

s = ½(v + v0)t

Substitute the v = v0 + at in above equation

s = ½[(v0 + at) + v0 ]t

s = ½(2v0 + at)t

•   Checking the slope for a p-t graph

The graphs of positions versus time for these two basic types of motion – constant velocity motion and accelerated motion (i.e., changing velocity) – demonstrate an important idea. According to the principle, the slope of a line on a position-time graph provides useful information about an object’s velocity. As the adage goes, “as the slope climbs, so does the velocity.”The slope will exhibit the same characteristics as the velocity. The slope is constant if the velocity is constant. If the velocity changes, the slope changes as well. The slope is positive if the velocity is positive. This same approach can be used for any type of motion.

•   Position- time graph

            Since the object began moving, a position-time graph displays how far it has gone from its initial position at any given time. The independent variable is usually time. It is stated to be dependent on other quantities, such as displacement. The location would be on the vertical axis (dependent variable) and time on the horizontal axis in a graph of position vs time (independent variable).

2. Speed

“The pace at which an object moves” is a scalar variable called speed. Speed is the rate at which an object travels over a given distance. A fast-moving object travels quickly and covers a large amount of ground in a short amount of time. A slow-moving object travelling at a low speed, on the other hand, covers a little amount of ground in the same amount of time. An item with zero speed is one that does not move at all.

• When time is constant, speed is proportional to distance.
• When the distance is constant, speed is inversely proportional to time.

    The pace at which a distance changes with time is referred to as speed.

    The average speed is the pace at which distance changes over time.

3.Velocity

“The rate at which an object changes its position,” says Velocity, a vector quantity. Consider a person who takes one stride forward and one step back at a rapid pace, always returning to the same starting place. While this would result in a flurry of activity, it would also result in a velocity of zero. Because the person continually returns to their former posture, the motion would never change position. This motion has zero velocity because velocity is defined as the location’s rate changes. If a person in motion wishes to accelerate, they must make every effort to increase the distance between them and their starting point. Every step must be taken to get that person further away from where he or she began. 

The quantity of velocity is a vector quantity. As a result, velocity knows where it is going.

Determining the direction of the velocity vector

The difficulty of describing the direction of the velocity vector is straightforward. The direction of the velocity vector is the same as the direction in which an object is moving. It makes no difference whether the thing is moving faster or slower. The velocity of an object traveling rightwards is known as rightwards velocity. If an object is going downwards, it is said to have a downward velocity. An airplane traveling west at 300 miles per hour has a velocity of 300 miles per hour, west. Note that speed has no direction (it is a scalar), while velocity is simply the speed number with a direction at any given time.

Conclusion

1. According to the principle, the slope of a line on a position-time graph provides useful information about an object’s velocity.
2. If the velocity is changing, then the slope is changing.
3. If velocity is positive, then the slope is positive.