Askiitians Tutor Team
Last Activity: 5 Days ago
To solve the problem of determining the minimum and maximum velocity \( v \) that a ball must have to hit only the third step after rolling off the edge of a roof, we need to analyze the physics of projectile motion and the geometry of the steps. Let's break this down step by step.
Understanding the Setup
We have a staircase where each step is 1 meter high and 1 meter wide. The ball rolls off the edge of the roof, which is at the height of the first step. The goal is for the ball to hit the third step only, meaning it must travel horizontally while falling vertically to land precisely on that step.
Key Parameters
- Height of each step: 1 meter
- Width of each step: 1 meter
- Steps to hit: 3rd step (2 meters high, 3 meters horizontally from the edge)
Projectile Motion Basics
When the ball rolls off the roof, it becomes a projectile. The motion can be analyzed in two components: horizontal and vertical. The horizontal motion is uniform, while the vertical motion is influenced by gravity.
Vertical Motion
The time \( t \) it takes for the ball to fall from the height of the roof to the height of the third step can be calculated using the equation of motion:
y = \frac{1}{2} g t^2
Here, \( y \) is the vertical distance (2 meters to the third step), and \( g \) is the acceleration due to gravity (approximately 9.81 m/s²). Rearranging gives:
t = \sqrt{\frac{2y}{g}} = \sqrt{\frac{2 \times 2}{9.81}} \approx 0.64 \text{ seconds}
Horizontal Motion
During this time, the ball must travel horizontally to reach the third step. The horizontal distance to the third step is 3 meters. The horizontal velocity \( v \) can be calculated using:
x = v \cdot t
Substituting the known values gives:
3 = v \cdot 0.64
From this, we can solve for \( v \):
v = \frac{3}{0.64} \approx 4.69 \text{ m/s}
Determining Minimum and Maximum Velocities
Now, we need to consider the conditions for the ball to hit only the third step. The minimum velocity occurs when the ball just reaches the edge of the third step without going further. The maximum velocity is when the ball just clears the edge of the third step without hitting the fourth step.
Minimum Velocity
The minimum velocity is the one calculated above, approximately 4.69 m/s. This is the speed required to ensure that the ball lands precisely on the third step.
Maximum Velocity
To find the maximum velocity, we need to ensure that the ball does not reach the fourth step. The fourth step is at a height of 3 meters. Using the same time of flight (0.64 seconds), we can calculate the horizontal distance it would travel at a higher speed:
x_{max} = v_{max} \cdot t
To avoid hitting the fourth step, the ball must not travel more than 3 meters horizontally in that time. Thus, we can set up the inequality:
v_{max} \cdot 0.64 < 4
Solving for \( v_{max} \):
v_{max} < \frac{4}{0.64} \approx 6.25 \text{ m/s}
Final Values
In summary, the minimum and maximum velocities for the ball to hit only the third step are:
- Minimum velocity: approximately 4.69 m/s
- Maximum velocity: less than approximately 6.25 m/s
This analysis shows how projectile motion principles and basic kinematics can be applied to solve real-world problems involving motion and gravity. Understanding these concepts helps in predicting the behavior of objects in motion.
