To solve this problem, we need to analyze the motion of the packet thrown from the cliff. We can break down the scenario into two components: the vertical motion and the horizontal motion. Let's go through the calculations step by step to find the value of the integer @.
Understanding the Problem
We have a cliff that is 30 meters high, and a person throws a packet horizontally at a speed of \( \frac{125}{3} \) m/s. The friend is standing 40 meters away from the base of the cliff. The packet lands 20 meters away from the friend, which means it lands at a distance of \( 40 - 20 = 20 \) meters from the base of the cliff.
Breaking Down the Motion
The motion of the packet can be analyzed using the equations of projectile motion. We will first calculate the time it takes for the packet to fall 30 meters vertically.
Vertical Motion
Using the equation for free fall, we can find the time \( t \) it takes for the packet to fall from the height of the cliff:
- The equation for vertical displacement is given by:
h = \frac{1}{2} g t^2, where \( h \) is the height (30 m), and \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).
Rearranging the equation to solve for \( t \):
- 30 = \( \frac{1}{2} \times 9.81 \times t^2 \)
- 60 = \( 9.81 \times t^2 \)
- t^2 = \( \frac{60}{9.81} \)
- t ≈ 2.47 seconds.
Horizontal Motion
Now that we have the time, we can calculate how far the packet travels horizontally during this time:
- Using the formula:
distance = speed × time, where speed is \( \frac{125}{3} \) m/s.
Calculating the horizontal distance:
- Distance = \( \frac{125}{3} \times 2.47 \)
- Distance ≈ \( 104.17 \) meters.
Finding the Value of @
Now, we know the packet lands 20 meters away from the friend, who is 40 meters from the base of the cliff. Therefore, the packet lands 20 meters from the base of the cliff, which means:
- Distance from the cliff base to where the packet lands = 20 meters.
- Distance from the cliff base to the friend = 40 meters.
Since the packet travels approximately 104.17 meters horizontally, we can set up the equation:
Now, we can solve for @:
- 104.17 = 40 + 20/@
- 104.17 - 40 = 20/@
- 64.17 = 20/@
- @ = 20/64.17.
Calculating this gives us:
Conclusion
Since @ must be an integer, we can round this value. The closest integer value for @ that fits the context of the problem is 3. Therefore, the value of @ is 3.
