Concept explainers
(i) The x component of the velocity versus time
(ii) The y component of the velocity versus time
(iii) The y component of the acceleration versus time.
Explanation of Solution
Given info:
Magnitude of the initial velocity
Angle made by vi with x- axis
Magnitude of the final velocity
Angle made by vf with x axis
Time interval during which the velocity changes
Formula used:
The components of the velocity vectors along the x and the y directions are given by,
The change in velocity along the x and the y directions is given by,
The components of the acceleration along the x and y directions are given by,
The equation to determine the variation of
The equation to determine the variation of
Calculation:
Resolve the vectors vi and vf along the x and the y axes. This is shown in the diagram below.
The components of the velocity vector vi are directed along +x and-y axes, hence the y component is assigned a negative sign. Calculate the magnitude of the components substituting the values of the variables in the equation,
The components of the final velocity vector vf are directed along the +x and +y directions, hence the components are positive. Calculate the magnitude of the components of the vector vf using the values of the variables.
Calculate the change in velocity along the x and the y directions.
Calculate the components of the acceleration along the x and the y directions.
Using the values of vix and ax in the equation
Plot a graph showing the variation of vx with time.
t in s | vxin m/s |
0 | 10 |
1 | 10.732 |
2 | 11.464 |
3 | 12.196 |
4 | 12.928 |
5 | 13.66 |
6 | 14.392 |
7 | 15.124 |
8 | 15.856 |
9 | 16.588 |
10 | 17.32 |
Using the values of viy and ay in the equation
Use the equation in a spread sheet and plot the graph showing the variation of vy with time t.
X-Values | Y-Values |
0 | -17.32 |
1 | -14.588 |
2 | -11.856 |
3 | -9.124 |
4 | -6.392 |
5 | -3.66 |
6 | -0.928 |
7 | 1.804 |
8 | 4.536 |
9 | 7.268 |
10 | 10 |
The y component of the acceleration ay is independent of time, since the object's acceleration remains constant during the time interval.
Plot a graph showing the variation of ay with time t.
t in s | ayin m/s2 |
0 | 2.732 |
1 | 2.732 |
2 | 2.732 |
3 | 2.732 |
4 | 2.732 |
5 | 2.732 |
6 | 2.732 |
7 | 2.732 |
8 | 2.732 |
9 | 2.732 |
10 | 2.732 |
Conclusion:
The graph showing the variation of vx with time is linear and the value increases from 10 m/s to 17.32 m/s at the end of 10 s.
The graph showing the variation of vy with time is also linear and its value increases from -17.32 m/s to reach a value of 10 m/s at the end of 10 s.
The graph showing the variation of ay with time is a straight line parallel to the time axis, showing that its value is invariant with time.
Thus all the results are consistent with an object moving with a constant acceleration during the time interval of 10s.
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Chapter 3 Solutions
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