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### Question 19: Kinetic Energy
To determine the kinetic energy of the pile driver, follow these steps:
1. Determine the weight of the pile driver:
- The weight [tex]\( W \)[/tex] is given as 8,000 Newtons.
2. Identify the velocity of the pile driver:
- The velocity [tex]\( v \)[/tex] is given as 6 m/s.
3. Understand the relationship between weight, mass, and gravity:
- The weight [tex]\( W \)[/tex] is related to the mass [tex]\( m \)[/tex] via the equation [tex]\( W = m \cdot g \)[/tex], where [tex]\( g \)[/tex] represents the acceleration due to gravity (9.8 m/s^2).
4. Calculate the mass:
[tex]\[ m = \frac{W}{g} = \frac{8000 \text{ N}}{9.8 \text{ m/s}^2} = 816.3265306122448 \text{ kg} \][/tex]
5. Use the kinetic energy formula:
- Kinetic energy [tex]\( KE \)[/tex] is given by [tex]\( KE = 0.5 \cdot m \cdot v^2 \)[/tex].
6. Substitute the values:
[tex]\[ KE = 0.5 \cdot 816.3265306122448 \text{ kg} \cdot (6 \text{ m/s})^2 = 14693.877551020407 \text{ Joules} \][/tex]
Thus, the kinetic energy of the pile driver is approximately 14,693.88 Joules.
### Question 20: Work Done
To determine the work done by the man pulling the sled, follow these steps:
1. Convert the distance from feet to meters:
- The distance [tex]\( d \)[/tex] is given as 20 feet.
- Use the conversion [tex]\( 1 \text{ foot} = 0.3048 \text{ meters} \)[/tex].
[tex]\[ d = 20 \text{ feet} \times 0.3048 \text{ m/foot} = 6.096 \text{ meters} \][/tex]
2. Convert the force from pounds to Newtons:
- The force [tex]\( F \)[/tex] is given as 200 pounds.
- Use the conversion [tex]\( 1 \text{ pound} = 4.44822 \text{ Newtons} \)[/tex].
[tex]\[ F = 200 \text{ pounds} \times 4.44822 \text{ N/pound} = 889.644 \text{ Newtons} \][/tex]
3. Convert the angle from degrees to radians:
- The angle [tex]\( \theta \)[/tex] is given as 40°.
- Use the conversion [tex]\( 1° = \frac{\pi}{180} \text{ radians} \)[/tex].
[tex]\[ \theta = 40° \times \frac{\pi}{180} = 0.6981317007977318 \text{ radians} \][/tex]
4. Use the work done formula:
- Work [tex]\( W \)[/tex] is given by [tex]\( W = F \cdot d \cdot \cos(\theta) \)[/tex].
5. Substitute the values:
[tex]\[ W = 889.644 \text{ N} \cdot 6.096 \text{ meters} \cdot \cos(0.6981317007977318) = 4154.465712210038 \text{ Joules} \][/tex]
Thus, the work done by the man pulling the sled is approximately 4,154.47 Joules.
To determine the kinetic energy of the pile driver, follow these steps:
1. Determine the weight of the pile driver:
- The weight [tex]\( W \)[/tex] is given as 8,000 Newtons.
2. Identify the velocity of the pile driver:
- The velocity [tex]\( v \)[/tex] is given as 6 m/s.
3. Understand the relationship between weight, mass, and gravity:
- The weight [tex]\( W \)[/tex] is related to the mass [tex]\( m \)[/tex] via the equation [tex]\( W = m \cdot g \)[/tex], where [tex]\( g \)[/tex] represents the acceleration due to gravity (9.8 m/s^2).
4. Calculate the mass:
[tex]\[ m = \frac{W}{g} = \frac{8000 \text{ N}}{9.8 \text{ m/s}^2} = 816.3265306122448 \text{ kg} \][/tex]
5. Use the kinetic energy formula:
- Kinetic energy [tex]\( KE \)[/tex] is given by [tex]\( KE = 0.5 \cdot m \cdot v^2 \)[/tex].
6. Substitute the values:
[tex]\[ KE = 0.5 \cdot 816.3265306122448 \text{ kg} \cdot (6 \text{ m/s})^2 = 14693.877551020407 \text{ Joules} \][/tex]
Thus, the kinetic energy of the pile driver is approximately 14,693.88 Joules.
### Question 20: Work Done
To determine the work done by the man pulling the sled, follow these steps:
1. Convert the distance from feet to meters:
- The distance [tex]\( d \)[/tex] is given as 20 feet.
- Use the conversion [tex]\( 1 \text{ foot} = 0.3048 \text{ meters} \)[/tex].
[tex]\[ d = 20 \text{ feet} \times 0.3048 \text{ m/foot} = 6.096 \text{ meters} \][/tex]
2. Convert the force from pounds to Newtons:
- The force [tex]\( F \)[/tex] is given as 200 pounds.
- Use the conversion [tex]\( 1 \text{ pound} = 4.44822 \text{ Newtons} \)[/tex].
[tex]\[ F = 200 \text{ pounds} \times 4.44822 \text{ N/pound} = 889.644 \text{ Newtons} \][/tex]
3. Convert the angle from degrees to radians:
- The angle [tex]\( \theta \)[/tex] is given as 40°.
- Use the conversion [tex]\( 1° = \frac{\pi}{180} \text{ radians} \)[/tex].
[tex]\[ \theta = 40° \times \frac{\pi}{180} = 0.6981317007977318 \text{ radians} \][/tex]
4. Use the work done formula:
- Work [tex]\( W \)[/tex] is given by [tex]\( W = F \cdot d \cdot \cos(\theta) \)[/tex].
5. Substitute the values:
[tex]\[ W = 889.644 \text{ N} \cdot 6.096 \text{ meters} \cdot \cos(0.6981317007977318) = 4154.465712210038 \text{ Joules} \][/tex]
Thus, the work done by the man pulling the sled is approximately 4,154.47 Joules.
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