Question

calculate the change in the kinetic energy (ke) of the bottle when the mass is increased. use the formula ( ke = \frac{1}{2}mv^2 ), where ( m ) is the mass and ( v ) is the speed (velocity). assume that the speed of the soda bottle falling from a height of 0.8 m will be 4 m/s, and use this speed for each calculation. record your calculations in table a of your study guide. when the mass of the bottle is 0.125 kg, the ke is dropdown kg·m²/s². when the mass of the bottle is 0.250 kg, the ke is... (dropdown options: 0.25, 0.5, 1, 2)

Explanation:

Step1: Identify formula and values

We use the kinetic energy formula \( KE = \frac{1}{2}mv^2 \), where \( m = 0.125 \, \text{kg} \) and \( v = 4 \, \text{m/s} \).

Step2: Substitute values into formula

Substitute \( m = 0.125 \) and \( v = 4 \) into \( KE = \frac{1}{2}mv^2 \):
\[
KE = \frac{1}{2} \times 0.125 \times (4)^2
\]

Step3: Calculate the expression

First, calculate \( (4)^2 = 16 \). Then:
\[
KE = \frac{1}{2} \times 0.125 \times 16
\]
\[
KE = 0.5 \times 0.125 \times 16
\]
\[
KE = 0.0625 \times 16
\]
\[
KE = 1
\]

Wait, there seems to be a miscalculation above. Let's recalculate:

Step2 (Correct): Substitute values into formula

\( KE = \frac{1}{2} \times 0.125 \times 4^2 \)

Step3 (Correct): Calculate step by step

First, \( 4^2 = 16 \). Then \( \frac{1}{2} \times 16 = 8 \). Then \( 8 \times 0.125 = 1 \)? Wait, no: \( 0.125 \times 8 = 1 \), yes. Wait, but let's check again. Wait, \( \frac{1}{2}mv^2 = 0.5 \times 0.125 \times 16 \). \( 0.5 \times 16 = 8 \), \( 8 \times 0.125 = 1 \). Wait, but maybe I made a mistake. Wait, no, \( 0.125 \) is \( \frac{1}{8} \), so \( 0.5 \times \frac{1}{8} \times 16 = 0.5 \times 2 = 1 \). Yes, that's correct. Wait, but the options include 0.25, 0.5, 1, 2. So for \( m = 0.125 \, \text{kg} \), \( KE = 1 \, \text{kg·m}^2/\text{s}^2 \)? Wait, no, wait: \( \frac{1}{2} \times 0.125 \times 16 = 0.5 \times 0.125 \times 16 = 0.0625 \times 16 = 1 \). Yes. Wait, but let's check with \( m = 0.250 \, \text{kg} \): \( \frac{1}{2} \times 0.250 \times 16 = 0.5 \times 0.250 \times 16 = 0.125 \times 16 = 2 \). Wait, maybe I misread the mass. Wait, the first mass is 0.125 kg. Let's recalculate:

\( KE = 0.5 \times 0.125 \times (4)^2 = 0.5 \times 0.125 \times 16 = 0.5 \times 2 = 1 \). Yes. So the answer for 0.125 kg is 1? Wait, but maybe the speed is different? Wait, the problem says "the speed of the soda bottle falling from a height of 0.8 m will be 4 m/s". So speed is 4 m/s. So for \( m = 0.125 \, \text{kg} \), \( KE = 0.5 * 0.125 * 16 = 1 \). So the correct value is 1.

Wait, but let's check again. \( 0.5 * 0.125 = 0.0625 \). \( 0.0625 * 16 = 1 \). Yes. So the answer is 1.

Answer:

1