Calculating Energy Lost Due to Air Friction

This problem involves applying the principle of conservation of energy to a scenario where air resistance causes some energy loss. We need to calculate the initial total energy of the body when it's thrown upwards and compare it to the potential energy it possesses at its maximum height. The difference will represent the energy lost due to air friction.

Understanding the Initial Conditions

• Mass of the body, $m = 1.5$ kg
• Initial velocity upwards, $v_i = 20$ m/s (from the top of the tower)
• Height of the tower, $h_{tower} = 10$ m
• Acceleration due to gravity, $g = 10$ m/s$^2$
• Maximum height reached from the ground, $H_{max} = 27$ m

Step 1: Calculate the Initial Energy

The initial energy ($E_i$) of the body consists of its kinetic energy ($KE_i$) because it's moving and its potential energy ($PE_i$) due to its position relative to the ground.

• Initial Kinetic Energy ($KE_i$): The formula for kinetic energy is $KE = \frac{1}{2} m v^2$. $KE_i = \frac{1}{2} \times 1.5 \, \text{kg} \times (20 \, \text{m/s})^2$ $KE_i = \frac{1}{2} \times 1.5 \times 400 \, \text{J}$ $KE_i = 0.75 \times 400 \, \text{J}$ $KE_i = 300 \, \text{J}$
• Initial Potential Energy ($PE_i$): The formula for potential energy is $PE = m g h$. Here, the height $h$ is the height of the tower. $PE_i = 1.5 \, \text{kg} \times 10 \, \text{m/s}^2 \times 10 \, \text{m}$ $PE_i = 15 \times 10 \, \text{J}$ $PE_i = 150 \, \text{J}
• Total Initial Energy ($E_i$): $E_i = KE_i + PE_i$ $E_i = 300 \, \text{J} + 150 \, \text{J}$ $E_i = 450 \, \text{J}

Step 2: Calculate the Final Energy at Maximum Height

At the maximum height ($H_{max}$) reached from the ground, the body momentarily stops before falling back down. Therefore, its velocity is zero, and it only possesses potential energy ($PE_f$).

• Final Potential Energy ($PE_f$): Using the maximum height from the ground: $PE_f = m g H_{max}$ $PE_f = 1.5 \, \text{kg} \times 10 \, \text{m/s}^2 \times 27 \, \text{m}$ $PE_f = 15 \times 27 \, \text{J}$ $PE_f = 405 \, \text{J}

Step 3: Determine the Energy Lost Due to Air Friction

The energy lost due to air friction ($E_{lost}$) is the difference between the total initial energy the body had and the total energy it has at its maximum height. In an ideal scenario (without friction), the initial energy would be equal to the potential energy at the maximum height. The discrepancy arises due to the energy dissipated by air resistance.

• Energy Lost ($E_{lost}$): $E_{lost} = E_i - PE_f$ $E_{lost} = 450 \, \text{J} - 405 \, \text{J}$ $E_{lost} = 45 \, \text{J}

Alternative Method: Focusing on Energy Above the Tower

We can also analyze the energy transformation starting from the point the body is thrown upwards.

• The initial kinetic energy provided is $KE_i = 300$ J.
• This kinetic energy is used to gain potential energy ($PE_{gain}$) as the body moves upwards and to overcome air friction ($E_{lost}$).
• The height the body reaches above the tower is $h_{up} = H_{max} - h_{tower} = 27 \, \text{m} - 10 \, \text{m} = 17 \, \text{m}$.
• The potential energy gained above the tower is $PE_{gain} = m g h_{up} = 1.5 \, \text{kg} \times 10 \, \text{m/s}^2 \times 17 \, \text{m} = 255 \, \text{J}$.
• Applying the energy balance for the upward motion above the tower: $KE_i = PE_{gain} + E_{lost}$.
• Therefore, the energy lost is $E_{lost} = KE_i - PE_{gain} = 300 \, \text{J} - 255 \, \text{J} = 45 \, \text{J}$.

Both methods confirm that the energy lost due to air friction is 45 Joules.