Question

v(t)= A (1-e^(-t/tmaxspeed))

1.Identify the

●units of the coefficient A

●physical meaning of A

●velocity of the car at t = 0

●Asymptote of this function as t → ∞?

2.Sketch a graph of velocity vs. time.

3.Derive an equation x(t) for the instantaneous position of the car as a function of time. Identify the

●value x when t = 0 s

●asymptote of this function as t → ∞

4.Sketch a graph of position vs. time.

5.Derive an equation for the instantaneous acceleration of the car as a function of time. Identify the

●acceleration of the car at t = 0 s

●asymptote of this function as t → ∞

6.Sketch a graph of acceleration vs. time.

7.Apply your mathematical models to your allocated car. Use the given data for the 0 – 28 m/s and 400m times to calculate the:

●value of the coefficient A

●maximum velocity

Maximum acceleration.

Answer 1

QUESTION 1

Since we know exactly the dimensions of some expressions that are included in the formula, we can conclude

$$\left\{\begin{array}{l} [v(t)]=\left[\displaystyle\frac{m}{sec}\right]\\[0.3cm] \left[1-e^{-t/t_{maxspeed}}\right]=[\text{just a number}] \end{array}\right.\rightarrow [A]=\left[\frac{m}{sec}\right]$$

We substitute $t=0$ and find the value of velocity :

$$v(0)=A\cdot\left(1-e^{-\displaystyle\frac{0}{t_{maxspeed}}}\right)=A\cdot(1-1)=0\\[0.3cm] \boxed{v(0)=0}$$

To find the asymptote as $t\to+\infty$ we calculate the limit :

$$v_{\infty}=\lim\limits_{t\to+\infty}\left(A\cdot\left(1-e^{-\displaystyle\frac{t}{t_{maxspeed}}}\right)\right)=A\cdot(1-0)=A\\[0.3cm] \boxed{v_\infty=A}$$

Now we can conclude about the physical meaning of the constant $A$ : $A$ is the boundary speed for a given type of motion.

ANSWER

$$\left\{\begin{array}{l} [A]=\left[\displaystyle\frac{m}{sec}\right]\\[0.3cm] A-\text{the boundary speed}\\[0.3cm] v(0)=0\\[0.3cm] v_\infty=A \end{array}\right.$$

QUESTION 2

To plot the graph, I chose the following constants:

$$A=10\\[0.3cm] t_{maxspeed}=5$$

QUESTION 3

As we know

$$v(t)=\frac{dx(t)}{dt}\to x(t)=\int v(t)dx=\int \left(A\cdot\left(1-e^{-\displaystyle\frac{t}{t_{maxspeed}}}\right)\right)dt\\[0.3cm] x(t)=A\cdot\left(t+t_{maxspeed}\cdot e^{-\displaystyle\frac{t}{t_{maxspeed}}}\right)+Const\\[0.3cm] \boxed{x(t)=A\cdot t_{maxspeed}\cdot\left(\frac{t}{t_{maxspeed}}+e^{-\displaystyle\frac{t}{t_{maxspeed}}}\right)+Const}$$

To find the starting position, we will assume that $Const=0$ , then

$$x(0)=A\cdot t_{maxspeed}\cdot\left(\frac{0}{t_{maxspeed}}+e^{-\displaystyle\frac{0}{t_{maxspeed}}}\right)\\[0.3cm] \boxed{x(0)=A\cdot t_{maxspeed}}$$

Using the same assumption $Const=0$ , we can find the asymptote for $t\to+\infty$ :

$$x_\infty=\lim\limits_{t\to+\infty}\left(A\cdot t_{maxspeed}\cdot\left(\frac{t}{t_{maxspeed}}+e^{-\displaystyle\frac{t}{t_{maxspeed}}}\right)\right)=+\infty\\[0.3cm] \boxed{x_\infty=+\infty}$$

ANSWER

General equation of motion:

$$x(t)=A\cdot t_{maxspeed}\cdot\left(\frac{t}{t_{maxspeed}}+e^{-\displaystyle\frac{t}{t_{maxspeed}}}\right)+Const$$

Under the assumption that $Const=0$ :

$$\left\{\begin{array}{l} x(0)=A\cdot t_{maxspeed}\\[0.3cm] x_\infty=+\infty \end{array}\right.$$

QUESTION 4

To plot the graph, I chose the following constants:

$$A=10\\[0.3cm] t_{maxspeed}=5$$

QUESTION 5

As we know

$$a(t)=\frac{dv(t)}{dt}=\frac{d}{dt}\left(A\cdot\left(1-e^{-\displaystyle\frac{t}{t_{maxspeed}}}\right)\right)\\[0.3cm] \boxed{a(t)=\frac{A}{t_{maxspeed}}\cdot e^{-\displaystyle\frac{t}{t_{maxspeed}}}}$$

Initial acceleration is

$$a(0)=\frac{A}{t_{maxspeed}}\cdot e^{-\displaystyle\frac{0}{t_{maxspeed}}}\\[0.3cm] \boxed{a(0)=\frac{A}{t_{maxspeed}}}$$

To find the asymptote as $t\to+\infty$ we calculate the limit :

$$a_\infty=\lim\limits_{t\to+\infty}\left(\frac{A}{t_{maxspeed}}\cdot e^{-\displaystyle\frac{t}{t_{maxspeed}}}\right)=0\\[0.3cm] \boxed{a_\infty=0}$$

ANSWER

$$\left\{\begin{array}{l} a(t)=\displaystyle\frac{A}{t_{maxspeed}}\cdot e^{-\displaystyle\frac{t}{t_{maxspeed}}}\\[0.3cm] a(0)=\displaystyle\frac{A}{t_{maxspeed}}\\[0.3cm] a_\infty=0 \end{array}\right.$$

QUESTION 6

To plot the graph, I chose the following constants:

$$A=10\\[0.3cm] t_{maxspeed}=5$$

QUESTION 7

Initial conditions $x(0)=400$ and $v(0)=28$ .

These initial conditions cannot be, since in point 2 we theoretically proved that the initial speed must be 0. Therefore, I can not answer this question. Moreover, you need to specify the value $t_{maxspeed}$ , although from the same paragraph 2 we can conclude that $t_{maxspeed}=+\infty$ , which is clearly not suitable for real tasks