Question

To achieve a speed of 2 m/s, the bottle must be dropped at m. To achieve a speed of 3 m/s, the bottle must be dropped at m. To achieve a speed of 4 m/s, the bottle must be dropped at m. To achieve a speed of 5 m/s, the bottle must be dropped at m. To achieve a speed of 6 m/s, the bottle must be dropped at m.

Answer 1

Answer:

$\begin{array}{l|l}\text{Speed}\; \mathrm{(m\cdot s^{-1})} & \text{Minimum Height\;(m)}\\\cline{1-2}\\[-1em] 2 & 0.204\\3&0.459\\4 & 0.815\\5 & 1.27 \\6 & 1.83\end{array}$.

Assumptions:

• The object is dropped in a free fall.
• There's no air resistance.
• The downward acceleration due to gravity is $\rm 9.81\;m\cdot s^{-2}$

Explanation:

Consider the "SUVAT" equation

$\displaystyle \frac{v^{2} - u^{2}}{2a} = x$,

where

• $v$ is the final velocity,
• $u$ is the initial velocity,
• $a$ is the acceleration of the object, and
• $x$ is the change in the object's position.

For example, if the bottle needs to achieve a speed of $v = \rm 2\; m\cdot s^{-1}$ by the time it reaches the ground,

• $u = 0$ since the statement that the bottle is "dropped" implies a free fall.
• $a = g = \rm 9.81\;m\cdot s^{-2}$.

Apply the previous equation to find the minimum height, $x$:

$\displaystyle x = \frac{v^{2} - u^{2}}{2a} = \rm \frac{\left(2\; m\cdot s^{-1}\right)^{2}}{2\times 9.81\; m\cdot s^{-2}} \approx 0.204\; m$.

Replace the $v$ value and apply the formula to find the minimum height required to reach different final speeds.

Answer 2

Answer:

0.20

0.46

0.82

1.28

1.84

Explanation: