Section 2.5 Motion in Space

2.5 Motion in Space

In this section, we see the applications of vector functions in the space. Recall in calculus I, for a given function $f(t)$ that presents the position of an object moving along a straight line with respect to the time $t$, we can find the velocity and the acceleration of this object by differentiate the function. At here, we are allowed to have the object moving in the plane or moving in the space. We just need to have two or three functions to present the position for each direction. 

Definition

Let $\overrightarrow{r(t)}$ be a twice-differentiable vector-valued function of the parameter $t$ that represents the position of an object as a function of time. The velocity vector $\overrightarrow{v(t)}$ of the object is given by  $$\text{Velocity}=\overrightarrow{v(t)}=\overrightarrow{r^{\prime }(t)}.$$ The acceleration vector $\overrightarrow{a(t)}$ is defined to be $$\text{Acceleration}=\overrightarrow{a(t)}=\overrightarrow{v^{\prime }(t)}=\overrightarrow{r”(t)}.$$ The speed is defined to be $$\text{Speed}=\parallel \overrightarrow{v(t)}\parallel =\parallel \overrightarrow{r^{\prime }(t)}\parallel =\frac{ds}{dt}.$$

Example 1: The position vector of an object moving in a plane is given by $\overrightarrow{r(t)}=<t^2,t^4-t>$. Find its velocity, speed, and acceleration when $t=1$. Sketch the curve along with the velocity vector at time $t=1$.

Exercise 1: The position vector of an object moving in a plane is given by $\overrightarrow{r(t)}=<t^3-t,t^2>$. Find its velocity, speed, and acceleration when $t=1$. Sketch the curve along with the velocity vector at time $t=1$.

Example 2: The position vector of an object moving in a plane is given by $\overrightarrow{r(t)}=<3t,\text{sin}(3t),\text{cos}(3t)>$. Find its velocity, speed, and acceleration when $t=0$. 

Exercise 2: The position vector of an object moving in a plane is given by $\overrightarrow{r(t)}=<\text{cos}(t),t,\text{sin}(t)>$. Find its velocity, speed, and acceleration when $t=0$. 

Recall in calculus I, when we drop an object from sky, we know the gravity gives the acceleration of the object because of Newton’s Second Law of Motion, $F=ma$. We can integrate the acceleration function to obtain the velocity and the position functions if the initial position and the initial velocity is known. This applies to the space as well.

Fact: Let $\overrightarrow{a(t)}$ be the acceleration vector function of a moving object in space with initial position $\overrightarrow{r(t_0)}$ and the initial velocity $\overrightarrow{v(t_0)}$ then

$$\overrightarrow{v(t)}=\overrightarrow{v(t_0)}+{\int }_{t_0}^t\overrightarrow{a(u)}du$$

$$\overrightarrow{r(t)}=\overrightarrow{r(t_0)}+{\int }_{t_0}^t\overrightarrow{v(u)}du.$$

Example 3: Find the velocity and the position vectors of a moving object with acceleration $\overrightarrow{a(t)}=<\text{cos}(2t),\text{sin}(2t),e^{2t}>$, initial velocity $\overrightarrow{v(0)}=<1,2,3>$ and initial position $\overrightarrow{r(0)}=<-2,0,2>$.

Exercise 3: Find the velocity and the position vectors of a moving object with acceleration $\overrightarrow{a(t)}=<e^t,\text{cos}(3t),\text{sin}(3t)>$, initial velocity $\overrightarrow{v(0)}=<1,2,3>$ and initial position $\overrightarrow{r(0)}=<-2,0,2>$.

Example 4: What force is required such that a particle of mass $m$ has position function $\overrightarrow{r(t)}=<t^2,t^3-t,t>$?

Exercise 4: What force is required such that a particle of mass $m$ has position function $\overrightarrow{r(t)}=<t-2,t^2-1,t^3>$?

Example 5: A projectile is fired with initial speed of $100$ m/s and angle of evaluation ${30}^0$. Find the range of the projectile, maximal heigh of the projectile and the impact of speed. 

Exercise 5: A projectile is fired with initial speed of $200$ m/s and angle of evaluation ${45}^0$. Find the range of the projectile, maximal heigh of the projectile and the impact of speed. 

Example 6: A ball is thrown at angle of ${30}^0$ to the air then it hits the ground that is $100$m away. What was the initial speed of the ball?

Group work:

1. Given that $\overrightarrow{r(t)}=<2\text{sin}(t),2\sqrt{2}\text{cos}(t),-2\text{sin}(t)>$ is the position vector of a moving particle, find the following quantities: The velocity of the particle. The speed of the particle. The acceleration of the particle. 

2. A ball is thrown at angle of ${60}^0$ to the air then it hits the ground that is $200$m away. What was the initial speed of the ball?

3. A golf ball is hit in a horizontal direction off the top edge of a building that is $100$ ft tall. How fast must the ball be launched to land $450$ft away?