You will drop the bottle/water mass so that it hits the lever at different speeds. Since an object in free fall is accelerated by gravity, you need to determine the heights necessary to drop the bottle to achieve the speeds of [tex]$2 m / s , 3 m / s , 4 m / s , 5 m / s$[/tex], and [tex]$6 m / s$[/tex]. Use the equation [tex]$H t=\frac{v^2}{2 g}$[/tex] to calculate the height, where [tex]$H t$[/tex] is the height, [tex]$v$[/tex] is the speed (velocity), and [tex]$g$[/tex] is the gravitational acceleration of [tex]$9.8 m / s ^2$[/tex]. Record these heights in Table B.
To achieve a speed of [tex]$2 m / s$[/tex], the bottle must be dropped at [$\square$] m.
To achieve a speed of [tex]$3 m / s$[/tex], the bottle must be dropped at [$\square$] m.
To achieve a speed of [tex]$4 m / s$[/tex], the bottle must be dropped at [$\square$] m.
To achieve a speed of [tex]$5 m / s$[/tex], the bottle must be dropped at [$\square$] m.
To achieve a speed of [tex]$6 m / s$[/tex], the bottle must be dropped at [$\square$] m.
Answer (1)
Calculate the height for 2 m/s: H t 1 = 2 × 9.8 2 2 ≈ 0.204 m .
Calculate the height for 3 m/s: H t 2 = 2 × 9.8 3 2 ≈ 0.459 m .
Calculate the height for 4 m/s: H t 3 = 2 × 9.8 4 2 ≈ 0.816 m .
Calculate the height for 5 m/s: H t 4 = 2 × 9.8 5 2 ≈ 1.276 m , and for 6 m/s: H t 5 = 2 × 9.8 6 2 ≈ 1.837 m . The heights are 0.204 m , 0.459 m , 0.816 m , 1.276 m , 1.837 m .
Explanation
Understanding the Problem We are given the formula H t = 2 g v 2 to calculate the height ( H t ) required to achieve a certain speed ( v ) under the influence of gravity ( g ). Here, g = 9.8 m / s 2 . We need to calculate the heights for v = 2 , 3 , 4 , 5 , and 6 m / s .
Calculating Height for 2 m/s First, let's calculate the height required to achieve a speed of 2 m / s :
H t 1 = 2 × 9.8 2 2 = 19.6 4 ≈ 0.204 m
Calculating Height for 3 m/s Next, let's calculate the height required to achieve a speed of 3 m / s :
H t 2 = 2 × 9.8 3 2 = 19.6 9 ≈ 0.459 m
Calculating Height for 4 m/s Now, let's calculate the height required to achieve a speed of 4 m / s :
H t 3 = 2 × 9.8 4 2 = 19.6 16 ≈ 0.816 m
Calculating Height for 5 m/s Then, let's calculate the height required to achieve a speed of 5 m / s :
H t 4 = 2 × 9.8 5 2 = 19.6 25 ≈ 1.276 m
Calculating Height for 6 m/s Finally, let's calculate the height required to achieve a speed of 6 m / s :
H t 5 = 2 × 9.8 6 2 = 19.6 36 ≈ 1.837 m
Final Answer Therefore, to achieve speeds of 2 m / s , 3 m / s , 4 m / s , 5 m / s , and 6 m / s , the bottle must be dropped from heights of approximately 0.204 m, 0.459 m, 0.816 m, 1.276 m, and 1.837 m, respectively.
Examples
Understanding the relationship between drop height and impact speed is crucial in various real-world applications. For example, engineers use this principle to design safety equipment like helmets or airbags, where controlling the impact force is essential. Similarly, in sports, athletes and coaches consider these factors to optimize performance and minimize the risk of injury. By calculating the necessary drop heights to achieve specific speeds, we can better understand and control the forces involved in collisions and impacts.
