Two spherical balls of the same radius but of different masses, are dropped at the same time from the top of a tower 19.6 m high. When they are 1.6 m above the ground, the balls will possess the same

Options

A. K.E.
B. P.E
C. momentum
D. total energy

Answer

To solve the problem, we need to analyze the situation of two spherical balls dropped from a height of 19.6 m and determine what they possess in common when they are 1.6 m above the ground.

### Step-by-Step Solution:

1. **Identify Initial Conditions:**
- Both balls are dropped from the same height (h = 19.6 m).
- The height above the ground when we are analyzing their properties is h' = 1.6 m.
- The initial velocity (u) of both balls is 0 m/s since they are dropped.

2. **Calculate the Distance Fallen:**
- The distance fallen (s) by each ball when they are at height h' is:
\[ s = h - h' = 19.6 \, \text{m} - 1.6 \, \text{m} = 18 \, \text{m} \]

3. **Use the Equation of Motion to Find Velocity:**
- We can use the third equation of motion to find the velocity (v) of each ball at height h':
\[ v^2 = u^2 + 2gs \]
- Substituting the values (u = 0, g = 9.8 m/s², s = 18 m):
\[ v^2 = 0 + 2 \cdot 9.8 \cdot 18 \]
\[ v^2 = 352.8 \implies v = \sqrt{352.8} \approx 18.8 \, \text{m/s} \]

4. **Kinetic Energy Calculation:**
- The kinetic energy (KE) of each ball can be calculated using the formula:
\[ KE = \frac{1}{2} mv^2 \]
- For ball 1 (mass m1):
\[ KE_1 = \frac{1}{2} m_1 v^2 \]
- For ball 2 (mass m2):
\[ KE_2 = \frac{1}{2} m_2 v^2 \]
- Since the velocities are the same, the kinetic energies will depend on their masses:
\[ KE_1 \neq KE_2 \quad \text{(because } m_1 \neq m_2\text{)} \]

5. **Potential Energy Calculation:**
- The potential energy (PE) of each ball at height h' is given by:
\[ PE = mgh' \]
- For ball 1:
\[ PE_1 = m_1 \cdot g \cdot h' \]
- For ball 2:
\[ PE_2 = m_2 \cdot g \cdot h' \]
- Again, since the masses are different, the potential energies will also be different:
\[ PE_1 \neq PE_2 \]

6. **Momentum Calculation:**
- The momentum (p) of each ball is given by:
\[ p = mv \]
- For ball 1:
\[ p_1 = m_1 \cdot v \]
- For ball 2:
\[ p_2 = m_2 \cdot v \]
- Since the masses are different, the momenta will also be different:
\[ p_1 \neq p_2 \]

7. **Total Energy Calculation:**
- The total mechanical energy (E) of each ball is the sum of its kinetic and potential energy:
\[ E = KE + PE \]
- For ball 1:
\[ E_1 = KE_1 + PE_1 \]
- For ball 2:
\[ E_2 = KE_2 + PE_2 \]
- However, since both balls were dropped from the same height and no energy is lost, the total energy for both balls remains constant and equal to their initial potential energy when dropped:
\[ E_1 = E_2 \]

### Conclusion:
When both balls are at the same height of 1.6 m above the ground, they possess the same total energy, even though their kinetic energy, potential energy, and momentum differ due to their different masses.