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Problem : A projectile is projected up with a velocity √(2ag) at an angle “θ” from an elevated position as shown in the figure. Find the maximum horizontal range that can be achieved.
Solution : In order to determine the maximum horizontal range, we need to find an expression involving horizontal range. We shall use the equation of projectile as we have the final coordinates of the motion as shown in the figure below :
$$y=x\mathrm{tan}\theta -\frac{g{x}^{2}}{2{u}^{2}{\mathrm{cos}}^{2}\theta}$$
Substituting and changing trigonometric ratio with the objective to create a quadratic equation in “tan θ” :
$$\Rightarrow -H=R\mathrm{tan}\theta -\frac{g{R}^{2}}{2\{\sqrt{\left(2ag\right)}{\}}^{2}}\left(1+{\mathrm{tan}}^{2}\theta \right)$$
Rearranging, we have :
$${R}^{2}{\mathrm{tan}}^{2}\theta -4aR\mathrm{tan}\theta +\left({R}^{2}-4aH\right)=0$$
$$\Rightarrow \mathrm{tan}\theta =\frac{4aR\pm \sqrt{\{{\left(4aR\right)}^{2}-4{R}^{2}\left({R}^{2}-4aH\right)\}}}{2{R}^{2}}$$
For tan θ to be real, it is required that
$$16{a}^{2}{R}^{2}\ge 4{R}^{2}\left({R}^{2}-4aH\right)$$
$$\Rightarrow 4{a}^{2}\ge \left({R}^{2}-4aH\right)$$
$$\Rightarrow {R}^{2}\le 4a\left(a+H\right)$$
$$\Rightarrow R\le \pm 2\sqrt{a\left(a+H\right)}$$
Hence, maximum possible range is :
$$\Rightarrow R=2\sqrt{a\left(a+H\right)}$$
Problem : A ball is thrown horizontally from the top of the tower to hit the ground at an angle of 45° in 2 s. Find the speed of the ball with which it was projected.
Solution : The question provides the angle at which the ball hits the ground. A hit at 45° means that horizontal and vertical speeds are equal.
$$\mathrm{tan}{45}^{0}=\frac{{v}_{y}}{{v}_{x}}=1$$
$$\Rightarrow {v}_{x}={v}_{y}$$
However, we know that horizontal component of velocity does not change with time. Hence, final velocity in horizontal direction is same as initial velocity in that direction.
$$\Rightarrow {v}_{x}={v}_{y}={u}_{x}$$
We can now find the vertical component of velocity at the time projectile hits the ground by considering motion in vertical direction. Here, ${u}_{y}=\mathrm{0,}\phantom{\rule{5pt}{0ex}}t=2s.$
Using equation of motion in vertical direction, assuming downward direction as positive :
$${v}_{y}={u}_{y}+at$$
$$\Rightarrow {v}_{y}=0+10X2=20\phantom{\rule{1em}{0ex}}m/s$$
Hence, the speed with which the ball was projected in horizontal direction is :
$$\Rightarrow {u}_{x}={v}_{y}=20\phantom{\rule{1em}{0ex}}m/s$$
Problem : A ball “A” is thrown from the edge of building “h”, at an angle of 30° from the horizontal, in upward direction. Another ball ”B” is thrown at the same speed from the same position, making same angle with horizontal, in vertically downward direction. If "u" be the speed of projection, then find their speed at the time of striking the ground.
Solution : The horizontal components of velocity for two projectiles are equal. Further, horizontal component of velocity remains unaltered during projectile motion. The speed of the projectile at the time of striking depends solely on vertical component of velocity. For the shake of convenience of analysis, we consider point of projection as the origin of coordinate system and vertically downward direction as positive y - direction.
Motion in vertical direction :
For A, $\phantom{\rule{2pt}{0ex}}{u}_{\mathrm{yA}}=-u\mathrm{sin}{30}^{0}=-\frac{u}{2}$
For B, $\phantom{\rule{2pt}{0ex}}{u}_{\mathrm{yB}}=u\mathrm{sin}{30}^{0}=\frac{u}{2}$
Thus, velocities in vertical direction are equal in magnitude, but opposite in direction. The ball,"A", which is thrown upward, returns after reaching the maximum vertical height. For consideration in vertical direction, the ball returns to the point of projection with same speed it was projected. What it means that the vertical component of velocity of ball "A" on return at the point projection is "u/2". This further means that two balls "A" and "B", as a matter of fact, travel down with same downward vertical component of velocity.
In other words, the ball "A" returns to its initial position acquiring same speed "u/2" as that of ball "B" before starting its downward journey. Thus, speeds of two balls are same i.e "u/2" for downward motion. Hence, two balls strike the ground with same speed. Let the final speed is "v".
Now, we apply equation of motion to determine the final speed in the vertical direction.
$${v}_{y}^{2}={u}_{y}^{2}+2gh$$
Putting values, we have :
$${v}_{y}^{2}={\left(\frac{u}{2}\right)}^{2}+2gh=\frac{\left({u}^{2}+8gh\right)}{4}$$
$${v}_{y}=\sqrt{\frac{\left({u}^{2}+8gh\right)}{2}}$$
The horizontal component of velocity remains same during the journey. It is given as :
$$vx=u\mathrm{cos}{30}^{0}=\frac{\sqrt{3}u}{2}$$
The resultant of two mutually perpendicular components is obtained, using Pythagoras theorem :
$$\Rightarrow {v}^{2}={v}_{x}^{2}+{v}_{y}^{2}=\frac{3{u}^{2}}{4}+\frac{\left({u}^{2}+8gh\right)}{4}$$
$$\Rightarrow {v}^{2}=\frac{\left(3{u}^{2}+{u}^{2}+8gh\right)}{4}={u}^{2}+2gh$$
$$\Rightarrow v=\sqrt{\left({u}^{2}+2gh\right)}$$
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