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# A missile fired from ground level rises $x$ metres vertically upwards in $t$ seconds and $x=100t-\large\frac{25}{2}t^{2}$ find the velocity with which the missile strikes the ground

Note: This is part 4th  of a 4 part question, split as 4 separate questions here.

Toolbox:
• If $s=f(t)$ is the distance function, representing the distance 's' travelled by a particle in time t, then the velocity and acceleration functions are $v=\large\frac{ds}{dt}$$=t'(t) and a=\large\frac{d^2s}{dt^2}=f''(t) • When a particle starts from rest,velocity v and time t are 0. When a particle is thrown up, it reaches maximum height at which v=0 and then falls back to earth. When a moving particle comes rest, v=0 • If y=f(x) then \large\frac{dy}{dx}$$=f'(x)$ is the rate of change of $y$ w.r.t $x$
• $\large\frac{dy}{dx_{(x_1,y_1)}}$ is the slope of the tangent to the curve at the point $(x_1,y_1)$ on the curve. It is the slope of the curve at that point.
• The normal at a point $(x_1,y_1)$ on $y=f(x)$ is perpendicular to the tangent at $(x_1,y_1)$
$x=100t-\large\frac{2s}{2}$$t^2 The velocity with which the missile strikes the ground when the missile strikes the ground x=0 Therefore 100t -\large\frac{25}{2}t^2=0 or \large\frac{t}{2}$$(200-25t)=0=>t=0,8$
Therefore the missile reaches the ground again at $t=8 \;secs$
Substitute $t=8$ in (ii) is obtain the velocity with which it strikes the ground.
The velocity $=100-25 \times 8$
$\qquad\quad=-100 \;m/sec$
(The negative sigh indicates the downward direction)