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In projectile motion, horizontal velocity is crucial in determining where an object lands. When the swimmer dives off the cliff, she does so with a horizontal leap. Horizontal velocity, in this context, refers to the constant speed at which she travels parallel to the ground.

Unlike vertical motion, which is affected by gravity, horizontal velocity remains steady because there are no forces like air resistance acting on it in an ideal scenario. To calculate it, we use the formula:

• \( d = vt \)

where \(d\) is the distance to clear (1.75 meters), \(v\) is the horizontal velocity, and \(t\) is the time spent in the air.

By knowing the time it takes to fall vertically, the swimmer's horizontal velocity can be calculated, allowing her to ensure she clears the ledge safely.

Free fall describes the motion of the swimmer as she descends from the cliff. This movement is under the sole influence of gravity, which means that the swimmer accelerates downwards at a constant rate. The acceleration due to gravity on Earth is \(9.81\, \text{m/s}^2\).

To find how long it takes for the swimmer to reach the ledge, we utilize the equation for free fall:

• \( y = \frac{1}{2} g t^2 \)

Here, \(y\) is the vertical displacement (9 meters in this case) and \(t\) is the time. Solving for \(t\) provides the time of fall, which is approximately 1.35 seconds.

This timing not only tells us when she will reach the ledge vertically but also plays a key role in calculating how far she travels horizontally during the same period.

Kinematics is the branch of physics that deals with the motion of objects. It provides the framework for analyzing this projectile motion problem. By breaking down the swimmer's dive into horizontal and vertical components, kinematics enables the separation of variables affecting each motion.

The key kinematic equations used here account for both aspects of her dive: the uniform horizontal motion and the accelerated vertical motion due to gravity.

Velocity in horizontal motion and displacement in free fall are connected through time. This makes kinematics essential for discovering the conditions under which the swimmer will land safely beyond the ledge.

Understanding kinematics helps in drawing clear relationships between the variables, such as distance, time, and velocity, thereby solving motion-related problems like this cliff dive example.

Gravity is the natural phenomenon that causes the swimmer to accelerate towards the Earth. It acts exclusively in the vertical direction in this scenario, leaving her horizontal motion unaffected.

The constant acceleration due to gravity is \(9.81\, \text{m/s}^2\). This value does not change and is what pulls the swimmer down as she leaps off the cliff.

In projectile motion, gravity is responsible for increasing the velocity of an object as it falls. The longer the object is in the air, the faster it travels downwards.

By understanding gravity, one can predict how fast and in what manner the swimmer descends. This not only assists in finding the time of fall but also helps in planning the necessary horizontal speed needed to ensure she doesn’t hit the ledge.