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velocity is v
0
= 0 ft/s, meaning you merely release the ball bearing and
don't throw it.
At this point, we don't even know what form x(t) should take, so we're
a bit stuck. The “front door” to this solution seems to be locked for us at
the moment, so instead we try to sneak around and enter through the back,
using an approach similar to the one we used earlier to define instantaneous
velocity. We'll consider ways that we might approximate the answer and
then watch what happens as the approximations get better and better.
Let's make our example a bit more specific. Earlier, we computed that
after being in free fall for 2.4 seconds, the ball bearing would have a velocity
of v(2.4) = 76.8 ft/s. However, we didn't say anything about how far it
had traveled during that time. Let's try to compute this distance, which is
x(2.4). To do this, we chop up the total 2.4 second interval into a number
of smaller “slices” of time, and approximate how far the ball bearing travels
during each slice. We can approximate the total distance traveled as the
sum of the distances traveled during each slice. To approximate how far the
ball bearing travels during one single slice, we first compute the velocity of
the ball bearing at the start of the slice by using Equation (11.13). Then we
approximate the distance traveled during the slice by plugging this velocity
as the constant velocity for the slice into Equation (11.14).
6 Slices, ∆t = 0.40
24 Slices, ∆t = 0.10
t
0
v
0
∆x
t
0
v
0
∆x
0.00
0.00
0.00
0.00
0.00
0.00
0.40
12.80
5.12
0.10
3.20
0.32
0.80
25.60
10.24
0.20
6.40
0.64
1.20
38.40
15.36
0.30
9.60
0.96
1.60
51.20
20.48
0.40
12.80
1.28
2.00
64.00
25.60
0.50
16.00
1.60
Total
76.80
0.60
19.20
1.92
0.70
22.40
2.24
0.80
25.60
2.56
0.90
28.80
2.88
12 Slices, ∆t = 0.20
1.00
32.00
3.20
1.10
35.20
3.52
t
0
v
0
∆x
1.20
38.40
3.84
0.00
0.00
0.00
1.30
41.60
4.16
0.20
6.40
1.28
1.40
44.80
4.48
0.40
12.80
2.56
1.50
48.00
4.80
0.60
19.20
3.84
0.80
25.60
5.12
1.60
51.20
5.12
1.00
32.00
6.40
1.70
54.40
5.44
1.20
38.40
7.68
1.80
57.60
5.76
1.40
44.80
8.96
1.90
60.80
6.08
1.60
51.20
10.24
2.00
64.00
6.40
1.80
57.60
11.52
2.10
67.20
6.72
2.00
64.00
12.80
2.20
70.40
7.04
2.20
70.40
14.08
2.30
73.60
7.36
Total
84.48
Total
88.32
Table 11.3.
Values for different numbers of slices
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