Sure, I'll provide a detailed, step-by-step solution to each of the questions.
### Question 1: Height of Cliff
Given:
- Mass of the rock ([tex]\( m \)[/tex]) = 24 kg
- Potential energy ([tex]\( PE \)[/tex]) = 3528 J
- Gravitational acceleration ([tex]\( g \)[/tex]) = 9.8 m/s²
We need to find the height of the cliff ([tex]\( h \)[/tex]) using the formula for potential energy:
[tex]\[ PE = mgh \][/tex]
Rearranging the formula to solve for height ([tex]\( h \)[/tex]) gives:
[tex]\[ h = \frac{PE}{mg} \][/tex]
Substitute the given values:
[tex]\[ h = \frac{3528 \, \text{J}}{24 \, \text{kg} \times 9.8 \, \text{m/s}^2} \][/tex]
Calculate:
[tex]\[ h = 14.999999999999998 \, \text{m} \][/tex]
So, the height of the cliff is approximately 15 meters.
### Question 2: Kinetic Energy of Cart
Given:
- Mass of the cart ([tex]\( m \)[/tex]) = 6 kg
- Velocity of the cart ([tex]\( v \)[/tex]) = 4 m/s
We need to find the kinetic energy ([tex]\( KE \)[/tex]) using the formula:
[tex]\[ KE = \frac{1}{2}mv^2 \][/tex]
Substitute the given values:
[tex]\[ KE = \frac{1}{2} \times 6 \, \text{kg} \times (4 \, \text{m/s})^2 \][/tex]
Calculate:
[tex]\[ KE = 0.5 \times 6 \times 16 \][/tex]
[tex]\[ KE = 48 \, \text{J} \][/tex]
So, the kinetic energy of the cart is 48 Joules.
### Question 3: Mass of Bicycle
Given:
- Velocity of the bicycle ([tex]\( v \)[/tex]) = 8 m/s
- Kinetic energy ([tex]\( KE \)[/tex]) = 384 J
We need to find the mass of the bicycle ([tex]\( m \)[/tex]) using the rearranged formula for kinetic energy:
[tex]\[ KE = \frac{1}{2}mv^2 \][/tex]
Rearranging to solve for mass ([tex]\( m \)[/tex]) gives:
[tex]\[ m = \frac{2KE}{v^2} \][/tex]
Substitute the given values:
[tex]\[ m = \frac{2 \times 384 \, \text{J}}{(8 \, \text{m/s})^2} \][/tex]
Calculate:
[tex]\[ m = \frac{768}{64} \][/tex]
[tex]\[ m = 12 \, \text{kg} \][/tex]
So, the mass of the bicycle is 12 kg.
In summary:
1. The height of the cliff is approximately 15 meters.
2. The kinetic energy of the cart is 48 Joules.
3. The mass of the bicycle is 12 kg.
