A missile fired from ground level rises $x$ metres vertically upward in $t$ second and $x = 100 t - \frac{25}{2} t^{2}$
Find
(i) the initial velocity of the missile.
(ii) the time when the height of the missile is a maximum
(iii) the maximum height reached and
(iv) the velocity with which the missile strikes the ground.

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Solution

Given upward displacement in time $t$ sec.
$x = 100 t \frac{25}{2} t^{2}$
(i) Initial velocity $= (\frac{d x}{d t}) t = 0 [i . e . \frac{d x}{d t} a t t = 0]$
$\frac{d x}{d t} = 100 - 25 t$
Initial velocity $= 100 - 25 \times 0 = 100$ m/s
(ii) At the maximum height velocity $\frac{d x}{d t} = 0$
$100 - 25 t = 0$
$\Rightarrow 25 t = 100$
$\Rightarrow t = 4$
This gives $t = 4$ sec.
(iii) Maximum height reached $=$ Height at $t = 4$ secs. $= x (4)$
$x = 100 \times 4 - \frac{25}{2} \times 16$
$= 400 - 200$
$= 200$ m
Hence the maximum height reached $= 200$ m.
(iv) When the missile reaches the ground the height $x = 0$ .
$\Rightarrow 100 t - \frac{25 t^{2}}{2} = 0$
This gives the value of $t = 0$ or $8$
Thus, $t = 0$ is not admissible
Taking $t = 8$ sec.
Velocity of the missile when it reaches the ground is
${(\frac{d x}{d t})}_{t = 8} = 100 - 25 \times 8 = - 100$ m/s.

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A missile fired from ground level rises x metres vertically upward in t second and x=100t−252t2Find (i) the initial velocity of the missile.(ii) the time when the height of the missile is a maximum(iii) the maximum height reached and(iv) the velocity with which the missile strikes the ground.