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

To achieve a speed of 2 m/s, the bottle must be dropped at

✔ 0.20

m.

To achieve a speed of 3 m/s, the bottle must be dropped at

✔ 0.46

m.

To achieve a speed of 4 m/s, the bottle must be dropped at

✔ 0.82

m.

To achieve a speed of 5 m/s, the bottle must be dropped at

✔ 1.28

m.

To achieve a speed of 6 m/s, the bottle must be dropped at

✔ 1.84

m.

Step-by-step explanation:

Answer 2

`\begin{array}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)`

Step-by-step explanation:

Consider the "SUVAT" equation

`\displaystyle (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 = (v^(2) - u^(2))/(2a) = \rm (\left(2\; m\cdot s^(-1)\right)^(2))/(2* 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.