A grasshopper can jump maximum distance 1.6 m. It spends negligible time on the ground. How far can it go in 10 seconds

The correct option is 20 √2 m.

Step-by-step explanation:
Let us analyze the motion of the grasshopper. Each jump of the grasshopper represents projectile motion. To cover the maximum horizontal distance in a given time, the grasshopper must jump at an angle that maximizes the range of each jump.

The maximum range of a projectile, $R_{\text{max}}$, is achieved when the projection angle is $\theta ={45}^{\circ }$.

The formula for the maximum range is given by:
$R_{\text{max}}=\frac{u^2}{g}$
where:
• $u$ is the initial speed of projection,
• $g$ is the acceleration due to gravity (taking $g=10{\text{m/s}}^2$).

Given that the maximum distance (range) of a single jump is $1.6\text{m}$:
$1.6=\frac{u^2}{10}$
$u^2=16\Rightarrow u=4\text{m/s}$

Now, let's calculate the time of flight $T$ for one single jump:
$T=\frac{2u\sin \theta }{g}$

Substituting $u=4\text{m/s}$, $\theta ={45}^{\circ }$, and $g=10{\text{m/s}}^2$:
$T=\frac{2\times 4\times \sin \left({45}^{\circ }\right)}{10}$
$T=\frac{8}{10}\times \frac{1}{\sqrt{2}}=\frac{4}{5\sqrt{2}}\text{seconds}$

Since the grasshopper spends negligible time on the ground, the total time of 10 seconds is entirely spent in the air during the jumps.
The number of jumps $n$ completed in $t=10\text{seconds}$ is:
$n=\frac{t}{T}=\frac{10}{\left(\frac{4}{5\sqrt{2}}\right)}$
$n=\frac{50\sqrt{2}}{4}=12.5\sqrt{2}\text{jumps}$

The total distance $d$ traveled by the grasshopper in 10 seconds is:
$d=n\times R_{\text{max}}$
$d=12.5\sqrt{2}\times 1.6$
$d=20\sqrt{2}\text{m}$