Chapter 6.3, Problem 58E

Solution

To find: the motion of the particle along with the horizontal line given by $s\left(t\right)$

When $t=-1$ , the particle is 10 units to the left of the y-axis at the point $\left(-10,5\right)$ ,

when $t=0$ , the particle is on the y-axis at the point $\left(0,5\right)$ ,

when $t=0.5$ , the particle is $7/8$ units to the right of the y-axis at the point $\left(\frac{7}{8},5\right)$ ,

when $t=1$ , the particle must change direction and it is on the y-axis at the point $\left(0,5\right)$ ,

when $t=2$ , the particle moves further and it is 4 units to the left of the y-axis at the point $\left(-4,5\right)$ ,

when $t=3$ , the particle is 6 units to the left of the y-axis at the point $\left(-6,5\right)$ ,

when $t=4$ , the particle must change the direction and it is on the y-axis,

when $t=5$ , the particle moves in the same direction and it is 20 units to the right of the y-axis at the point $\left(20,5\right)$ .

Given information:

The given expression is $s\left(t\right)=t^3-5t^2+4t$ , $-1\le t\le 5$ .

Formula used:

Calculation:

Choose arbitrarily the horizontal line $y=5$ to display the motion. The graph $C_1$ of the parametric equations,

  $C_1:x_1=t^3-5t^2+4t,$

  $y_1=5$ , $-1\le t\le 5$ ,

simulates the motion. The particle at any time $t$ is given by the point $\left(x_1\left(t\right),y_1\right)$ .

Substitute the value of $t$ in $x_1=t^3-5t^2+4t,$

then the following can be observed:

when $t=-1$ , the particle is 10 units to the left of the y-axis at the point $\left(-10,5\right)$ ,

when $t=0$ , the particle is on the y-axis at the point $\left(0,5\right)$ ,

when $t=0.5$ , the particle is $7/8$ units to the right of the y-axis at the point $\left(\frac{7}{8},5\right)$ ,

when $t=1$ , the particle must change direction and it is on the y-axis at the point $\left(0,5\right)$ ,

when $t=2$ , the particle moves further and it is 4 units to the left of the y-axis at the point $\left(-4,5\right)$ ,

when $t=3$ , the particle is 6 units to the left of the y-axis at the point $\left(-6,5\right)$ ,

when $t=4$ , the particle must change the direction and it is on the y-axis,

when $t=5$ , the particle moves in the same direction and it is 20 units to the right of the y-axis at the point $\left(20,5\right)$ .

The above observations can be traced through below figures:

  

  $t=-1$ ; $x=-10$ , $y=5$ ;

Figure (1)

  

  $t=0.5$ ; $x=0.875$ , $y=5$ ;

Figure (2)

  

  $t=1$ ; $x=0$ , $y=5$ ;

Figure (3)

  

   $t=2$ ; $x=-4$ , $y=5$ ;

Figure (4)

  

  $t=5$ ; 20, $y=5$ ;

Figure (5)