Answer :
To find out how much more kinetic energy the bowling ball has when rolling at 16 mph compared to 14 mph, we use the formula for kinetic energy:
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
where:
- [tex]\( m \)[/tex] is the mass of the object (in kilograms),
- [tex]\( v \)[/tex] is the speed of the object (in meters per second).
Step 1: Calculate the kinetic energy at 16 mph (7.1 meters per second):
1. Use the formula with the mass [tex]\( m = 6 \)[/tex] kg and speed [tex]\( v = 7.1 \)[/tex] m/s.
2. Plug these values into the formula:
[tex]\[ KE_1 = \frac{1}{2} \times 6 \times (7.1)^2 \][/tex]
3. Calculate the value:
[tex]\[ KE_1 = 0.5 \times 6 \times 50.41 \][/tex]
[tex]\[ KE_1 = 151.23 \, \text{Joules} \][/tex]
Step 2: Calculate the kinetic energy at 14 mph (6.2 meters per second):
1. Use the same formula with speed [tex]\( v = 6.2 \)[/tex] m/s.
2. Plug these values into the formula:
[tex]\[ KE_2 = \frac{1}{2} \times 6 \times (6.2)^2 \][/tex]
3. Calculate the value:
[tex]\[ KE_2 = 0.5 \times 6 \times 38.44 \][/tex]
[tex]\[ KE_2 = 115.32 \, \text{Joules} \][/tex]
Step 3: Determine the difference in kinetic energy:
1. Subtract the kinetic energy at 14 mph from the kinetic energy at 16 mph:
[tex]\[ \text{Difference in } KE = KE_1 - KE_2 \][/tex]
2. Substitute the calculated values:
[tex]\[ \text{Difference in } KE = 151.23 - 115.32 \][/tex]
3. Calculate the difference:
[tex]\[ \text{Difference in } KE = 35.91 \, \text{Joules} \][/tex]
Therefore, the bowling ball has 35.9 Joules more kinetic energy when rolling at 16 mph compared to when it is rolling at 14 mph.
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
where:
- [tex]\( m \)[/tex] is the mass of the object (in kilograms),
- [tex]\( v \)[/tex] is the speed of the object (in meters per second).
Step 1: Calculate the kinetic energy at 16 mph (7.1 meters per second):
1. Use the formula with the mass [tex]\( m = 6 \)[/tex] kg and speed [tex]\( v = 7.1 \)[/tex] m/s.
2. Plug these values into the formula:
[tex]\[ KE_1 = \frac{1}{2} \times 6 \times (7.1)^2 \][/tex]
3. Calculate the value:
[tex]\[ KE_1 = 0.5 \times 6 \times 50.41 \][/tex]
[tex]\[ KE_1 = 151.23 \, \text{Joules} \][/tex]
Step 2: Calculate the kinetic energy at 14 mph (6.2 meters per second):
1. Use the same formula with speed [tex]\( v = 6.2 \)[/tex] m/s.
2. Plug these values into the formula:
[tex]\[ KE_2 = \frac{1}{2} \times 6 \times (6.2)^2 \][/tex]
3. Calculate the value:
[tex]\[ KE_2 = 0.5 \times 6 \times 38.44 \][/tex]
[tex]\[ KE_2 = 115.32 \, \text{Joules} \][/tex]
Step 3: Determine the difference in kinetic energy:
1. Subtract the kinetic energy at 14 mph from the kinetic energy at 16 mph:
[tex]\[ \text{Difference in } KE = KE_1 - KE_2 \][/tex]
2. Substitute the calculated values:
[tex]\[ \text{Difference in } KE = 151.23 - 115.32 \][/tex]
3. Calculate the difference:
[tex]\[ \text{Difference in } KE = 35.91 \, \text{Joules} \][/tex]
Therefore, the bowling ball has 35.9 Joules more kinetic energy when rolling at 16 mph compared to when it is rolling at 14 mph.