Chapter 6.3, Problem 56E

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 4 units to the left of the y-axis at the point $\left(-4,5\right)$ ,

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

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

when $t=3$ , the particle must change the direction and it is 3 units to the right of the y-axis at the point $\left(3,5\right)$ ,

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

Given information:

The given expression is $s\left(t\right)=-t^2+4t,$

  $-1\le t\le 5$ .

Formula used:

Calculation:

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

  $C_1:x_1=-t^2+4t,$

  $y_1=5$ , $-1\le t\le 5$ ,

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 $x_1=-t^2+4t$, then the following can be observed:

when $t=-1$ , the particle is 4 units to the left of the y-axis at the point $\left(-4,5\right)$ ,

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

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

when $t=3$ , the particle must change the direction and it is 3 units to the right of the y-axis at the point $\left(3,5\right)$ ,

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

The above observations can be traced through below figures:

  

  $t=-1$ ; $x=-4$ , $y=5$ ;

Figure (1)

  

  $t=1$ ; $x=3$ , $y=5$ ;

Figure (2)

  

  $t=2$ ; $x=4$ , $y=5$ ;

Figure (3)

  

  $t=5$ ; $x=-5$ , $y=5$ ;

Figure (4)