Question

Show all of your work in the space provided. Show all of your work to get full credit. All answers have to be in SI units(20 pints for each questions).

2.Following are the two sets of position and time data collected using a motion sensor and cart.

       Table 1. table 2

Time (s)	Position (m)	V(instantaneous) (m/s)
0	10.0
0.25	13.0
0.50	16.5
0.75	21.0
1.00	26.5
1.25	32.3
1.50	38.0
1.75	44.5
2.00	52.0
2.25	60.0
2.50	68.5
2.75	77.5
3.00	87.0
3.25	97.0
3.50	108.0
3.75	119.5
4.00	131.5

Time (s)	Position (m)	V(instantaneous) (m/s)
1	0.972
1.25	1.139
1.5	1.269
1.75	1.362
2	1.416
2.25	1.432
2.5	1.412
2.75	1.356
3	1.265
3.25	1.139
3.5	0.979

  

1. For Table 1 and Table 2 data plot on graph paper the position vs time graph and identify the type of motion of the cart on the track whether it is on horizontal plane or inclined plane. If the best fit coefficients (quadratic coefficients) for Table 1 data are A=4.6,    B=11.8,C= 9.8 and for Table 2 data are A= -0.29, B=1.31, C= -0.049 write the corresponding equation of motion connecting position velocity and acceleration in each cases. Identify the initial position, initial velocity and acceleration in both cases.(6 point)

      

2. Calculate the instantaneous velocities in both cases(fill in tables) and plot the graph of instantaneous velocity vs time for Table 1 and Table 2 data. While calculating the instantaneous velocity, use time interval of 0.25s.(4 point)
3. Identify the best fit equations in the above cases and compare with known equation of motion and obtain the values of initial velocity and acceleration. How these values compare with values in part A? What is the average velocity in both cases?.(6 point)

Answer 1

Excel chart tabulated for position and velocity as a function of time is given below.

This table and the following figures and equations are made for data set-1

Position-time graph for table-1 is given below

Using best-fit of second degree polynomial, the following equation for position as a function of time is found

x(t) = 4.639 t2 + 11.83 t + 9.791

velocity ( dx/dt ) = 9.378 t + 11.83

acceleration ( d2 x / dt2 ) = 9.378

Instantaneous Velocity-time graph is given below

From best fit , function for instantaneous velocity is obtained as

v(t) = 9.409 + 10.38

Average velocity = Distance / time = 131.5 m / 4s = 32.875 m/s

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Excel chart tabulated for position and velocity as a function of time is given below.

This table and the following figures and equations are made for data set-2

Position-time graph is shown below

Using best-fit of second degree polynomial, the following equation for position as a function of time is found

x(t) = -0.29 t2 + 1.31 t -0.049

velocity ( dx/dt ) = -0.58 t + 1.31

acceleration ( d2 x / dt2 ) = -0.58

Instantaneous Velocity-time graph is given below

From best fit , function for instantaneous velocity is obtained as

v(t) = -0.583 t + 1.389

Average velocity = Distance / time = 0.913 m/ 2.5 s = 0.365 m/s

( in this case forward distance travelled is 0.46 m and backward distance travelled is 0.453 m

Total distance = 0.46+0.453 = 0.913 m in 2.5 s )