Two identical balls are thrown directly upward, ball \( A \) at speed _ V _ and ball_ B_ at speed 2 _ v , and they feel no air resistance. Which statement about these balls is correct?

To analyze the motion of the two balls, we can consider their initial velocities and the effects of gravity on their trajectories. 1. **Ball A** is thrown upward with an initial speed \( V \). 2. **Ball B** is thrown upward with an initial speed \( 2V \). Both balls will experience the same gravitational acceleration downward, which is approximately \( 9.81 \, \text{m/s}^2 \) (assuming we are near the Earth's surface). ### Key Points to Consider: - **Time to Reach Maximum Height**: The time it takes for each ball to reach its maximum height can be calculated using the formula: \[ t = \frac{v}{g} \] where \( v \) is the initial velocity and \( g \) is the acceleration due to gravity. - For Ball A: \[ t_A = \frac{V}{g} \] - For Ball B: \[ t_B = \frac{2V}{g} = 2 \cdot \frac{V}{g} = 2t_A \] This means Ball B takes twice as long to reach its maximum height compared to Ball A. - **Maximum Height**: The maximum height reached by each ball can be calculated using the formula: \[ h = \frac{v^2}{2g} \] - For Ball A: \[ h_A = \frac{V^2}{2g} \] - For Ball B: \[ h_B = \frac{(2V)^2}{2g} = \frac{4V^2}{2g} = \frac{2V^2}{g} \] This shows that Ball B reaches a maximum height that is four times greater than that of Ball A. ### Conclusion: - Ball B, thrown with a speed of \( 2V \), reaches a greater maximum height and takes longer to reach that height compared to Ball A, which is thrown with speed \( V \). Thus, the correct statement about these balls is that **Ball B reaches a higher maximum height and takes longer to reach that height than Ball A**.