To determine how much more kinetic energy a 6-kilogram bowling ball has when rolling at 16 mph compared to when rolling at 14 mph, we will first calculate the kinetic energy at both speeds using the kinetic energy formula:
\[ KE = \frac{1}{2}mv^2 \]
Step 1: Convert speeds to meters per second
- 16 mph = 7.1 m/s (as given)
- 14 mph = 6.2 m/s (as given)
Step 2: Calculate the kinetic energy at 16 mph
Using \( m = 6 \) kg and \( v = 7.1 \) m/s:
\[ KE_{16} = \frac{1}{2} \times 6 , \text{kg} \times (7.1 , \text{m/s})^2 \] \[ KE_{16} = 3 \times (7.1^2) \] \[ KE_{16} = 3 \times 50.41 , \text{m}^2/\text{s}^2 \] \[ KE_{16} = 151.23 , \text{J} \]
Step 3: Calculate the kinetic energy at 14 mph
Using \( m = 6 \) kg and \( v = 6.2 \) m/s:
\[ KE_{14} = \frac{1}{2} \times 6 , \text{kg} \times (6.2 , \text{m/s})^2 \] \[ KE_{14} = 3 \times (6.2^2) \] \[ KE_{14} = 3 \times 38.44 , \text{m}^2/\text{s}^2 \] \[ KE_{14} = 115.32 , \text{J} \]
Step 4: Calculate the difference in kinetic energy
\[ \Delta KE = KE_{16} - KE_{14} \] \[ \Delta KE = 151.23 , \text{J} - 115.32 , \text{J} \] \[ \Delta KE = 35.91 , \text{J} \]
Rounding this to one decimal place gives us approximately 35.9 J.
Final Answer
The difference in kinetic energy is 35.9 J.