91Ó°ÊÓ

In projectile motion, the initial velocity is the speed at which an object begins its journey. In this scenario, the golf ball has an initial velocity of 11.4 meters per second, but it is solely in the horizontal direction.
This happens because the ball rolls off the cliff without any initial push in the vertical direction. Thus, it has no initial vertical velocity and only begins to gain vertical speed due to gravity.

• The initial velocity affects how far the ball will travel horizontally.
• It helps in determining the overall path or trajectory of the projectile.

In other cases, the initial velocity could have both horizontal and vertical components, but here, the horizontal component is the only one present at the start.

The vertical motion of a projectile is affected by the acceleration due to gravity, which is approximately 9.8 meters per second squared on Earth. As the golf ball falls, its vertical motion is independent of its horizontal motion.
To find how long the ball is in the air, we use the kinematic equation for vertical motion:

• There is no initial vertical velocity.
• The ball falls due to gravity, which accelerates it downward.

The kinematic equation for vertical displacement is:
\[ y = v_{y0} t + \frac{1}{2} g t^2 \]
Where:

• \(y\) is the vertical displacement (15.5 m in this case).
• \(v_{y0}\) is the initial vertical velocity.
• \(g\) is the acceleration due to gravity.
• \(t\) is the time in the air.

In this equation,
When we plug in the values from this particular problem, we find that the time spent in the air is about 1.78 seconds.

Horizontal motion in projectile problems like this one is straightforward because it involves constant speed. With no horizontal acceleration acting on the golf ball after it leaves the cliff, its horizontal velocity remains constant at 11.4 m/s throughout the flight.

• The horizontal motion does not influence how long the ball spends in the air.
• While vertical motion is affected by gravity, horizontal motion continues unfettered.

The key concept here is that in projectile motion, the horizontal velocity doesn't change unless there's an additional force present. Since we're considering only gravity acting upon the ball, the horizontal component stays constant until impact.

Kinematic equations describe the motion of objects and they are pivotal in solving problems about projectile motion. They help us dissect the complex pathways that objects like our golf ball take, by looking at different components separately.The key kinematic equations cover:

• Time, given initial velocities and acceleration.
• Displacements in both the vertical and horizontal directions.
• Final velocities based on initial velocities and accelerations.

In our problem, the vertical motion is analyzed using:
\[ y = v_{y0} t + \frac{1}{2} g t^2 \]
And to find the final vertical speed:
\[ v_{yf} = v_{y0} + gt \]
This can be vital when calculating the speed just before impact.
Finally, for the overall speed of the ball as it strikes the water, we used:
\[ v = \sqrt{v_{x}^2 + v_{yf}^2} \]
These equations let us calculate and predict the behavior of projectiles by splitting their motion into solvable parts. The horizontal and vertical motions are treated independently, simplifying the problem-solving process.