Problem 74 - College Physics

Question

\(\bullet$$\bullet\) A 65.0 -kg parachutist falling vertically at a speed of 6.30 \(\mathrm{m} / \mathrm{s}\) impacts the ground, which brings him to a complete stop in a distance of 0.92 \(\mathrm{m}\) (roughly half of his height). Assuming constant acceleration after his feet first touch the ground, what is the average force exerted on the parachutist by the ground?

Step-by-Step Solution

Answer

The average force exerted by the ground is approximately 1406 N upward.

Step 1: Identify Given Values

We are given the following values: - Mass of the parachutist, \( m = 65.0 \) kg - Initial speed, \( v_i = 6.30 \) m/s - Final speed, \( v_f = 0 \) m/s (as he comes to a complete stop) - Distance over which he stops, \( d = 0.92 \) m.

Step 2: Use Kinetic Equation to Find Acceleration

To find the acceleration, we use the kinematic equation:\[ v_f^2 = v_i^2 + 2a d \]Substitute the known values:\[ 0 = (6.30)^2 + 2a \times 0.92 \]Solve for \( a \):\[ a = \frac{- (6.30)^2}{2 \times 0.92} \approx -21.63 \, \mathrm{m/s^2} \]

Step 3: Calculate the Force Using Newton's Second Law

Newton's second law states \( F = ma \). We have: - \( m = 65.0 \) kg - \( a = -21.63 \) m/s²Substitute these into the equation:\[ F = 65.0 \times (-21.63) \approx -1405.95 \, \mathrm{N} \]

Step 4: Interpret the Negative Sign

The negative sign indicates that the force acting on the parachutist is in the opposite direction to his motion. Hence, the average force exerted by the ground on the parachutist is approximately 1406 N upward.

Key Concepts

Kinematics

Kinematics deals with the motion of objects without considering the forces that cause the motion. It's essentially the "geometry" of motion. In this problem, the kinematics concept helps us understand how the parachutist decelerates from a speed to a stop over a certain distance. When the parachutist touches the ground, his motion is described by kinematic equations that help us find the deceleration. These equations link velocity, acceleration, and displacement, providing a complete picture of his trajectory without directly involving the forces. The kinematic equation used here, \[ v_f^2 = v_i^2 + 2a d \] helps us determine how the initial speed and stopping distance can be used to calculate the acceleration as the parachutist comes to a stop. This foundational understanding of motion is critical in analyzing and predicting physical phenomena.

Newton's Second Law

Newton's Second Law is a cornerstone in physics that connects motion with force. It states that the acceleration of an object is dependent on two variables: the net force acting on the object and its mass. This relationship is expressed as \( F = ma \).Newton's Second Law is crucial in our problem, where we use it to calculate the average force exerted by the ground on the parachutist. By knowing the mass and the acceleration calculated via kinematic equations, we apply the law to find the stopping force. This force is calculated to be around 1406 N, which acts upwards against the falling parachutist.The negative sign in acceleration and force indicates the direction of force is opposite to the motion. This concept ensures students learn how forces relate to changes in motion, a fundamental part of solving real-world physics problems.

Acceleration

Acceleration is the rate at which an object changes its velocity. When the parachutist hits the ground, he experiences a rapid deceleration or negative acceleration, bringing him to a stop.Using the kinematic equation, we calculated acceleration as \( a = -21.63 \, \mathrm{m/s^2} \).This negative value signs a decrease in speed, typical when an object slows down. In physics problems, understanding how and when acceleration acts helps in predicting motion patterns. It's particularly important here because it directly affects the force exerted on the parachutist, influencing how safely he lands. Acceleration is thus a crucial concept connecting kinematics and dynamics.

Kinematic Equations

Kinematic equations serve as tools for predicting an object's motion when its acceleration is constant. These equations relate initial velocity, final velocity, acceleration, time, and displacement.In our problem, we used the equation \[ v_f^2 = v_i^2 + 2a d \], which connects the initial speed, final speed (zero here), acceleration, and stopping distance. Solving for acceleration gave insight into how quickly the parachutist decelerates.Kinematic equations are essential for analyzing uniform acceleration scenarios, such as a parachutist slowing down upon impact. They provide the mathematical framework to solve for unknown variables, making them indispensable tools in physics problem-solving. This understanding is vital for students, offering a methodical approach to dissecting physics problems.