EXAMPLE 3 Given x as a function of t. A jet engine moves along an exper- imental track (which we call the x axis) as shown in Fig. 13a . We will treat the engine as if it were a particle. Its position as a function of time is given by the equation x=At^2+B where A=2.10m/s^2 and B=2.80m , and this equa- tion is plotted in Fig. 13b. (a)Determine the displacement of the engine during the time interval from t_(1)=3.00 s to t_(2)=5.00s (b) Determine the average velocity during this time interval. (c)Determine the magnitude of the instanta- neous velocity at t=5.00s

(a) To determine the displacement of the engine during the time interval from $t_{1}=3.00$ s to $t_{2}=5.00s$, we need to find the difference in position at these two times. We can do this by plugging in the values of $t_{1}$ and $t_{2}$ into the equation $x=At^{2}+B$ and subtracting the two results.<br /><br />Using $t_{1}=3.00$ s:<br />$x_{1}=A(t_{1})^{2}+B=2.10(3.00)^{2}+2.80=20.10$ m<br /><br />Using $t_{2}=5.00$ s:<br />$x_{2}=A(t_{2})^{2}+B=2.10(5.00)^{2}+2.80=52.80$ m<br /><br />The displacement is then:<br />$\Delta x=x_{2}-x_{1}=52.80-20.10=32.70$ m<br /><br />(b) To determine the average velocity during this time interval, we can use the formula:<br />$\text{average velocity}=\frac{\Delta x}{\Delta t}$<br /><br />where $\Delta t=t_{2}-t_{1}=5.00-3.00=2.00$ s<br /><br />Plugging in the values:<br />$\text{average velocity}=\frac{32.70}{2.00}=16.35$ m/s<br /><br />(c) To determine the magnitude of the instantaneous velocity at $t=5.00$ s, we need to find the derivative of the position function $x=At^{2}+B$ with respect to time $t$ and evaluate it at $t=5.00$ s.<br /><br />The derivative of $x$ with respect to $t$ is:<br />$\frac{dx}{dt}=2At$<br /><br />Plugging in $t=5.00$ s:<br />$v=\frac{dx}{dt}\bigg|_{t=5.00}=2A(5.00)=2(2.10)(5.00)=21.00$ m/s<br /><br />Therefore, the magnitude of the instantaneous velocity at $t=5.00$ s is $21.00$ m/s.