91Ó°ÊÓ

In the realm of physics, free fall is a specific type of motion where an object experiences acceleration due to gravity alone. When an object is in free fall, it does not encounter any other forces, such as air resistance. The primary force acting upon it is the pull of gravity, resulting in a vertical acceleration directed downwards.

For earth-bound objects, the acceleration due to gravity is approximately 9.81 m/s². Using this constant, we can predict how quickly an object will fall over time, which is crucial for solving problems like the one in the exercise. In our scenario, the swimmer falls 9 meters vertically, and we use the equation for free fall: \[ h = \frac{1}{2} g t^2 \]where:

• h is the height (9.00 m in this case)

• g is the acceleration due to gravity (9.81 m/s²)
• t is the time in seconds

Rearranging the equation to solve for time (t), we calculate how long the swimmer is in the air before potentially hitting the ledge. The process involves substituting the known values and solving for t, leading to a time of approximately 1.354 seconds.

Horizontal velocity refers to the constant speed at which an object travels horizontally. In projectile motion, while vertical velocity changes due to gravity, the horizontal component remains steady, assuming negligible air resistance. This distinction is important when analyzing a horizontally launched projectile, like the swimmer in the exercise.

To determine the minimum horizontal velocity, one must ensure that the swimmer covers the horizontal distance (1.75 m) in the time calculated for the free fall. This calculation uses the formula:\[ x = v_x t \]

• x represents the horizontal distance to be cleared (1.75 m)

• v_x is the horizontal velocity
• t is the time in seconds (1.354 s)

Using this relation, we find that the swimmer needs a horizontal velocity of approximately 1.29 m/s to clear the ledge. This speed effectively stretches the horizontal travel over the time descended, ensuring a safe leap from the cliff.

Kinematic equations are pivotal in solving problems involving motion, especially when detailing projectile paths. They provide mathematical tools to describe an object's motion using variables such as initial velocity, final velocity, time, and acceleration. The primary kinematic equation used here is:\[ d = v_i t + \frac{1}{2} a t^2 \]

• d stands for the distance covered

• v_i is the initial velocity
• a represents acceleration (gravity for vertical motion)
• t is the time under observation

In the current exercise, these equations allow for the separate analysis of vertical (free fall) and horizontal (constant velocity) motions. The separation simplifies solving complex motion because it acknowledges that horizontal and vertical components of motion don’t influence each other in a projectile. This approach streamlines calculating how the swimmer clears the ledge. It underscores that in projectiles, one must consider two motions together to solve the problem effectively.