Q.
Given questions are based on the given information. There are four options for following questions out of which only one is correct. The motion of a car is represented by the equation.
$x(t)=0.08 t^{3}$
Where, $x(t)$ is position of the car at time instant 't' There is a table given below providing the value of $\Delta x / \Delta t$ calculated for $\Delta t$ equals $2.0 s , 1.0 s , 0.5 s$
$0.1 s$ and $0.01 s$ centred at $t =4.0 s$. The second and
the third columns give the value of $t_{1}=\left(t-\frac{\Delta t}{2}\right)$ and $t_{2}=\left(t+\frac{\Delta t}{2}\right)$ and the fourth and fifth columns give the
corresponding values of ' $x$ '.
$\Delta t (s)$
$t_{1}(s)$
$t_{2}(s)$
$x(t) (m)$
$\Delta x (m)$
$\Delta x/ \Delta t (ms^{-1})$
2.0
3.0
5.0
2.16
7.84
3.92
1.0
3.5
4.5
3.43
B
3.86
0.5
3.75
A
4.21875
1.9225
3.84
0.1
3.95
4.05
4.93039
0.38402
3.8402
0.01
3.995
4.005
5.100824
0.0384
C
The value of $\frac{d x}{d t}$ at $t=4.0\, s$
| $\Delta t (s)$ | $t_{1}(s)$ | $t_{2}(s)$ | $x(t) (m)$ | $\Delta x (m)$ | $\Delta x/ \Delta t (ms^{-1})$ |
|---|---|---|---|---|---|
| 2.0 | 3.0 | 5.0 | 2.16 | 7.84 | 3.92 |
| 1.0 | 3.5 | 4.5 | 3.43 | B | 3.86 |
| 0.5 | 3.75 | A | 4.21875 | 1.9225 | 3.84 |
| 0.1 | 3.95 | 4.05 | 4.93039 | 0.38402 | 3.8402 |
| 0.01 | 3.995 | 4.005 | 5.100824 | 0.0384 | C |
Motion in a Straight Line
Solution: