WAEC 2009 · Paper 2 · Q8

The position vector of a body, with respect to the origin, is given by $\mathbf r = 4t\,\mathbf i + (12 - 3t)\,\mathbf j$ at any time $t$ seconds.

(a)
Find the velocity of the body.
Separate values with commas, e.g. 3, −2
(b)
Calculate the magnitude of the displacement between $t = 0$ and $t = 5$ .

Worked solution (try it first)

(a)

Velocity is the rate of change of position: $\mathbf v = \dfrac{d\mathbf r}{dt}$ .
Differentiate each component: $\dfrac{d}{dt}(4t) = 4$ and $\dfrac{d}{dt}(12 - 3t) = -3$ .
So $\mathbf v = 4\mathbf i - 3\mathbf j$ m s $^{-1}$ (a constant velocity).

(b)

At $t = 0$ : $\mathbf r = 0\mathbf i + 12\mathbf j$ .
At $t = 5$ : $\mathbf r = 20\mathbf i + (12 - 15)\mathbf j$
$= 20\mathbf i - 3\mathbf j$ .
Displacement $= (20\mathbf i - 3\mathbf j) - 12\mathbf j$
$= 20\mathbf i - 15\mathbf j$ .
Magnitude $= \sqrt{20^2 + 15^2} = \sqrt{625} = 25$ units.

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