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Make a qualitative graph of speed v versus time t for a falling object (a) for which air resistance can be ignored and (b) for which air resistance cannot be ignored.

Make a qualitative graph of speed v versus time t for a falling object (a) for which air resistance can be ignored and (b) for which air resistance cannot be ignored.

Grade:10

2 Answers

Jitender Pal
askIITians Faculty 365 Points
6 years ago
We will depict the speed versus time t graph by deriving the relation between the final velocity of the object and time. And then we will modify the relation to make it similar to the mathematical equation to plot it on the speed-time graph
We neglect the effect of air resistance and consider the downward motion of the object, and assume that the initial speed of the object to be v0y while the final speed to be vy .
We take the convention that the vectors pointing downwards are taken to be positive.
From the equation of kinematics, we have
vy = v0y + gt
It is important to realize that the initial velocity and the acceleration due to gravity points in the same direction, therefore the final velocity or the speed increases over time, therefore the final speed can be represented as:
vy = v0y + gt …… (1)
The equation above resembles the equation of straight line of form y = ax + b , where a and b are constants.
Comparing the equation (1) with the standard equation of straight line, we have that the time is represented on x xis while the final speed on y axis such that the constant a and b are g and v0y respectively.
The plot of above equation is shown below:
235-2195_18.PNG
(b) Now we do not neglect the effect of air resistance and consider the downward motion of the object, and assume that the initial speed of the object to be v0y while the final speed to be vy.
We also assume that the net acceleration of the object is given by , the mass by m and the force of friction by Fair.
From the equation of kinematics, we have
vy = v0y + at
It is important to note that here we cannot consider the acceleration of the object to be because of the resistance it experiences from air.
The magnitude of the acceleration can be calculated as:
a = \frac{mg-F_{air}}{m}
One can clearly note that if the magnitude of Fair is zero, the value of acceleration is equal to the acceleration due to gravity .
If we assume that the magnitude of Fairalways remain constant during the journey, the magnitude of a will be a constant value but smaller than the magnitude of g .
On substituting the value of in the equation of kinematics, we have
v_{y} = v_{0y} + \left \left ( \frac{mg- F_{air}}{m} \right )t
Therefore we get a straight line similar to that in above part but the change of speed will be smaller in magnitude relatively as can be seen in the graph below.
The graph between the speed and time in such situation is given below:
235-809_19.PNG
It is important to note that in nature, the magnitude of Fair changes with the speed of the object, that is for the higher speed the magnitude of Fair is larger and therefore the smaller net acceleration. If we consider a linear relationship between the speed and Fair, the decrease in acceleration will be constant. However if we consider the non-linear relationship between the speed and Fair , the change in acceleration will very fall very quickly such that the final speed will tend to become constant after certain time.
The speed-time graph in such a case would resemble like the one shown below:
235-1771_20.PNG
Jitender Pal
askIITians Faculty 365 Points
6 years ago
We will depict the speed versus time t graph by deriving the relation between the final velocity of the object and time. And then we will modify the relation to make it similar to the mathematical equation to plot it on the speed-time graph
We neglect the effect of air resistance and consider the downward motion of the object, and assume that the initial speed of the object to be v0y while the final speed to be vy .
We take the convention that the vectors pointing downwards are taken to be positive.
From the equation of kinematics, we have
vy = v0y + gt
It is important to realize that the initial velocity and the acceleration due to gravity points in the same direction, therefore the final velocity or the speed increases over time, therefore the final speed can be represented as:
vy = v0y + gt …… (1)
The equation above resembles the equation of straight line of form y = ax + b , where a and b are constants.
Comparing the equation (1) with the standard equation of straight line, we have that the time is represented on x xis while the final speed on y axis such that the constant a and b are g and v0y respectively.
The plot of above equation is shown below:
235-2195_18.PNG
(b) Now we do not neglect the effect of air resistance and consider the downward motion of the object, and assume that the initial speed of the object to be v0y while the final speed to be vy.
We also assume that the net acceleration of the object is given by , the mass by m and the force of friction by Fair.
From the equation of kinematics, we have
vy = v0y + at
It is important to note that here we cannot consider the acceleration of the object to be because of the resistance it experiences from air.
The magnitude of the acceleration can be calculated as:
a = \frac{mg-F_{air}}{m}
One can clearly note that if the magnitude of Fair is zero, the value of acceleration is equal to the acceleration due to gravity .
If we assume that the magnitude of Fairalways remain constant during the journey, the magnitude of a will be a constant value but smaller than the magnitude of g .
On substituting the value of in the equation of kinematics, we have
v_{y} = v_{0y} + \left \left ( \frac{mg- F_{air}}{m} \right )t
Therefore we get a straight line similar to that in above part but the change of speed will be smaller in magnitude relatively as can be seen in the graph below.
The graph between the speed and time in such situation is given below:
235-809_19.PNG
It is important to note that in nature, the magnitude of Fair changes with the speed of the object, that is for the higher speed the magnitude of Fair is larger and therefore the smaller net acceleration. If we consider a linear relationship between the speed and Fair, the decrease in acceleration will be constant. However if we consider the non-linear relationship between the speed and Fair , the change in acceleration will very fall very quickly such that the final speed will tend to become constant after certain time.
The speed-time graph in such a case would resemble like the one shown below:
235-1771_20.PNG

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