7. Exercises

A football ball is kicked so that its position, in metres, at time t seconds is given by

\displaystyle \mathbf{r}=6t\mathbf{i}+\left( 15t-4\textrm{.}9{{t}^{2}} \right)\mathbf{j}

where \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are horizontal and vertical unit vectors respectively.

a) Find the position of the ball when t = 0, 1, 2, 3 and 4 seconds.

b) Plot the path of the ball.

c) From your plot estimate the horizontal distance travelled by the ball when it hits the ground.

Two children, A and B, run so that their position vectors in metres at time \displaystyle t seconds are given by:

\displaystyle {{\mathbf{r}}_{A}}=t\mathbf{i}+\frac{9t}{2}\mathbf{j} and \displaystyle {{\mathbf{r}}_{B}}=t\mathbf{i}+(9t-{{t}^{2}})\mathbf{j}

Plot the paths of the two children for \displaystyle 0\le t\le 4.5 . What happens when t=4.5?

A motor boat moves so that its position, in metres, at time t seconds is given by

\displaystyle \mathbf{r}=(2t-16)\mathbf{i}+(4t-12)\mathbf{j}

where \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are unit vectors that are directed east and north respectively.

A rock has position vector \displaystyle 174\mathbf{i}+352\mathbf{j}.

a) Find the position of the boat, when t = 0, 40, 80 and 120 seconds.

b) Plot the path of the boat.

c) Find the time when the boat is due north of the rock.

d) Find the time when the boat is due east of the rock.

When a frisby moves so that its position, in metres, at time t seconds is modelled by

\displaystyle \mathbf{r}=(5t-{{t}^{\ 2}})\mathbf{i}+7t\mathbf{j}+( 8t-{{t}^{\ 2}} )\mathbf{k}

where \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are perpendicular horizontal unit vectors and \displaystyle \mathbf{k} is a vertical unit vector, and the origin is at ground level.

a) Find the time when the frisby hits the ground.

b) Find the position of the frisby when it hits the ground.

c) Find the distance between the point where the frisby was thrown and the point where it lands.