91Ó°ÊÓ

The angle of elevation is crucial in projectile motion as it determines the initial direction of the projectile. In this scenario, it refers to the angle at which the water is being shot from the hose. This is a critical variable because it influences how far and how high the water can travel. To find the angle of elevation, you can use the relationship between horizontal distance, initial velocity, and time.
For the water to reach the burning building 45 meters away, the angle of elevation must be set so that the horizontal component of velocity can cover this distance in the given time of 3 seconds. By applying the formula \ \( d_x = v_0 \cos \alpha \times t \ \), and knowing that \ \( d_x \) is 45 m, \ \( v_0 \) is 25 m/s, and \ \( t \) is 3 s, we can calculate the angle \( \alpha \) to be approximately 53.13 degrees. This ensures the water stream reaches the target effectively.

Kinematic equations are a set of equations that describe the motion of objects without considering the forces that cause this motion. They are essential in solving projectile motion problems. These equations relate the five key kinematic variables: displacement, initial velocity, final velocity, acceleration, and time.
In our exercise, they help determine the trajectory and final outcomes of the water stream after it is projected. For the vertical rise and fall of the water, the kinematic equation \ \( y = v_{0y} \times t - \frac{1}{2} g t^2 \ \) helps calculate how high the stream goes above the ground. Similarly, we use \ \( v = v_{0y} - g \times t \ \) to calculate the vertical component of velocity before the water hits the building. Understanding how these equations apply is essential, as they enable us to predict aspects like the highest point of travel and the impact speed against the building.

In projectile motion, it's beneficial to break the initial velocity into horizontal and vertical components. This simplification makes calculations manageable and helps understand the trajectory better.
The horizontal component, calculated as \ \( v_{0x} = v_0 \cos \alpha \ \), influences how far the projectile travels horizontally. For our water stream, this determines whether it reaches the building. The vertical component, \ \( v_{0y} = v_0 \sin \alpha \ \), is responsible for the ascent and descent of the projectile. It dictates how high the water will rise before gravity pulls it back down. At the highest point, the vertical velocity is zero, meaning the water momentarily stops moving upwards before descending.
Splitting these components helps solve for different variables in projectile motion, such as how to find the height the water hits the building, or its speed at any given point.

Speed refers to how fast something is moving, while acceleration describes the rate at which this speed changes. During projectile motion, speed and acceleration vary at different points along the path.
At the highest point of a projectile's path, the vertical component of the velocity becomes zero, but the horizontal component remains unchanged because there's no horizontal acceleration (neglecting air resistance). Therefore, the speed at this point is purely horizontal. For our water stream, this speed is 15 m/s purely in the horizontal direction.
Acceleration is constant throughout at 9.81 m/s² downwards (due to gravity). This affects the vertical velocity of the water, causing it to decrease as it rises and then increases in magnitude (but in the downward direction) as it falls back down. Understanding these changes helps us determine critical outcomes, like how fast and at what angle the water will hit the building.