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Welcome to our Physics lesson on Mathematics of Position vs Time Graph, this is the third lesson of our suite of physics lessons covering the topic of Position v's Time and Distance v's Time Graph, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
In the previous tutorial "Equations of Motion", one of the motion equations in the uniformly accelerated (decelerated) motion was
Since ∆x⃗ = x⃗ - x⃗0, we obtain
Or,
We can also write the above equation in such a way that the powers of the independent variable decrease from left to right, i.e.
Time (t) represents the independent variable in this equation while the final position x⃗ is the dependent variable. This means this is a quadratic function (as stated in the previous tutorail) because the highest order of the independent variable is two (time is at power two in the second term of the equation).
The mathematical formula of a quadratic function is
In the actual application of the above equation in Kinematics, we have
From Mathematics, it is known that the graph of a quadratic function is a parabola. When the coefficient A is positive, the arms of the parabola are upwards, and when A is negative, the parabola is arm-down. This means when acceleration is positive, the arms of the parabola are upwards, i.e. the position increases more and more for equal time intervals as seen in the example of the previous paragraph. Look at the graphs below:
If we have a negative (arms-down) parabola, the first half of the graph means the object still moves in the positive direction but it is slowing down, until it stops. Then, it turns back, i.e. it starts moving towards negative (although it may still be in the positive part of the position). This motion now is accelerated although the sign of acceleration is negative (this occurs due to the fact that the object is moving towards negative as discussed in the Remark paragraph of the previous tutorial).
This is the reason why we take the initial sign of acceleration to describe the entire motion although we may know the object changes direction as in the case of an object thrown upwards at velocity v⃗0.
The table below clarifies the sign of acceleration in different situations:
How the object is moving? | Speeding up towards positive | Slowing down towards positive | Speeding up towards negative | Slowing down towards negative |
---|---|---|---|---|
What is the sign of acceleration? | Positive | Negative | Negative | Positive |
Let's consider a numerical example.
The position vs time graph below belongs to an object thrown upwards. Calculate the initial velocity v⃗0 of this object and the total time of flight t.
We have the following information from the graph:
Thus, using the above info, we obtain
We take g⃗ = -10 m/s2 as the object initially is moving up. Therefore, substituting the values, we obtain
To find the total time of flight, we use the above result to calculate the time needed for the object to reach the highest point, i.e. first we have to find tup. Then, we have to multiply the result by 2 (by symmetry) to get the total time of flight.
Thus, applying the first equation of vertical motion we have
Hence:
We can also find the same result by applying directly the quadratic equation of vertical motion. Thus, the object is in two instants at the height h⃗ = 0: one when it starts moving up (at t1 = 0) and the other when it falls again on the ground after flight (at t2). Hence, we have
Substituting the known values, we obtain
We have
and
As you see, both methods give the same results.
You have reach the end of Physics lesson 3.9.3 Mathematics of Position vs Time Graph. There are 4 lessons in this physics tutorial covering Position v's Time and Distance v's Time Graph, you can access all the lessons from this tutorial below.
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