Free Fall Motion and Gravitational Acceleration

Gravitational acceleration near Earth's surface is approximately 9.8 m/s², a value derived from the gravitational force exerted by Earth on objects. This constant is crucial for calculations in physics and engineering, as it affects the motion of falling objects and the trajectory of projectiles. Variations in this value can occur due to altitude and geographical location, but 9.8 m/s² is the accepted average used in most calculations. Understanding this standard is essential for accurately predicting how objects will behave under the influence of gravity.

In free fall motion, the displacement of a falling object is considered negative because it moves downward, typically defined as the negative direction in a coordinate system. This convention aligns with the standard Cartesian coordinate system, where upward movement is positive and downward movement is negative. Therefore, as the object falls towards the ground, its displacement is measured as a negative value relative to its starting position.

To find the velocity of the ball just before it hits the ground, we use the formula \( v = g \cdot t \), where \( g \) is the acceleration due to gravity (9.8 m/s²) and \( t \) is the time in seconds (2.75 s). Calculating this gives \( v = 9.8 \times 2.75 = 26.95 \) m/s. The negative sign indicates the direction of the velocity is downward, which is consistent with the ball falling. Thus, the final velocity just before impact is -26.95 m/s.

To determine the time it takes for a ball to fall a displacement of -37.06 m under the influence of gravity, we can use the equation of motion: \( s = ut + \frac{1}{2}gt^2 \). Here, the initial velocity (u) is 0 m/s, \( g \) is 9.8 m/s² (acceleration due to gravity), and \( s \) is -37.06 m. Plugging in the values and solving the quadratic equation for time yields approximately 2.75 seconds, which is the time taken for the ball to hit the ground.

As an object falls freely under the influence of gravity, it accelerates downward at a constant rate, typically 9.81 m/s² near the Earth's surface. This means that its velocity increases in magnitude as it falls, as it gains speed with each passing second. The force of gravity continuously acts on the object, causing it to accelerate rather than maintain a constant velocity or decrease in speed. Therefore, the correct description of its velocity is that it increases as the object falls.

In physics, the convention is to define upward as the positive direction. Since gravitational acceleration acts downward towards the center of the Earth, it is assigned a negative value. This helps maintain consistency in equations of motion, where a negative value indicates that the acceleration is in the opposite direction to the defined positive direction. Thus, when an object is in free fall, its acceleration due to gravity is represented as a negative value to reflect this downward direction.

When a ball is dropped from rest, it starts with no initial motion. Therefore, its initial velocity is zero. The acceleration due to gravity (9.8 m/s²) will act on the ball after it is released, causing it to gain speed as it falls, but at the moment of release, the velocity is still 0 m/s.

To find the displacement of a ball dropped from rest after 3 seconds, we can use the kinematic equation d = ½gt². Here, g is the acceleration due to gravity, which is -9.8 m/s². Plugging in the values, we calculate d as follows: d = ½ × (-9.8 m/s²) × (3 s)² = ½ × (-9.8) × 9 = -44.1 m. The negative sign indicates that the displacement is downward, consistent with the direction of gravity. Thus, the displacement after 3 seconds is -44.1 m.

To find the velocity of the ball just before it hits the ground, we can use the kinematic equation: \( v^2 = u^2 + 2gh \). Here, the initial velocity \( u \) is 0 m/s (since the ball is dropped), \( g \) is 9.8 m/s², and \( h \) is 37.06 m. Plugging in the values, we get \( v^2 = 0 + 2 \times 9.8 \times 37.06 \). Solving for \( v \) gives approximately -26.95 m/s, indicating the ball is moving downward just before impact. The negative sign reflects the direction of the velocity.

This equation, v = v₀ + gt, directly relates the final velocity (v) of an object to its initial velocity (v₀), the acceleration due to gravity (g), and the time (t) it has been falling. For a freely falling object, the initial velocity is often zero, simplifying the calculation. This equation is particularly useful for determining how fast the object is moving after a specific time under the influence of gravity, making it the most straightforward choice for this scenario.