Problem 40 - College Physics

Question

Human biomechanics. World-class sprinters can spring out of the starting blocks with an acceleration that is essentially horizontal and of magnitude 15 \(\mathrm{m} / \mathrm{s}^{2}\) . (a) How much horizontal force must a 55 -kg sprinter exert on the starting blocks during a start to produce this acceleration? (b) What exerts the force that propels the sprinter, the blocks or the sprinter himself?

Step-by-Step Solution

Answer

(a) 825 N; (b) The blocks exert the force that propels the sprinter.

Step 1: Identify the Known Variables

Determine the values given in the problem. You are provided with the sprinter's mass \( m = 55 \, \text{kg} \) and the horizontal acceleration \( a = 15 \, \text{m/s}^2 \).

Step 2: Use Newton's Second Law

To find the force, apply Newton's second law of motion. The formula is: \[ F = ma \] where \( F \) is the force, \( m \) is the mass, and \( a \) is the acceleration.

Step 3: Calculate the Force

Substitute the known values into the formula: \[ F = 55 \, \text{kg} \times 15 \, \text{m/s}^2 \] \[ F = 825 \, \text{N} \] Thus, the sprinter must exert a force of 825 N on the blocks.

Step 4: Determine What Exerts the Force

According to Newton's third law, for every action there is an equal and opposite reaction. The force exerted by the sprinter on the blocks is matched by the blocks exerting the same amount of force on the sprinter. Thus, the force that propels the sprinter forward is exerted by the blocks.

Key Concepts

Newton's Second Law

Newton's Second Law is a cornerstone in the study of physics and is crucial for understanding how forces affect motion. It states that the force acting on an object is equal to the mass of that object multiplied by its acceleration. The formula for this law is written as \[ F = ma \]where:

• \( F \) represents the force applied in newtons (N),
• \( m \) represents the mass in kilograms (kg),
• \( a \) stands for acceleration in meters per second squared (m/s²).

This equation tells us two things:- More force is required to move heavier objects.- Increased acceleration requires more force on a given mass. Applying Newton's Second Law helps predict how an object will be propelled under known forces. In the context of a sprinter, using this law allows us to calculate the force needed to achieve a specific acceleration using their known mass.

Force Calculation

Calculating the force involves understanding the relationship between mass, acceleration, and force defined by Newton’s Second Law. Given the mass of a sprinter is 55 kg and their required acceleration is 15 m/s², we use the formula \( F = ma \)to find that \[ F = 55 \, \text{kg} \times 15 \, \text{m/s}^2 \]\[ F = 825 \, \text{N} \]This means that a sprinter weighing 55 kg needs to exert a force of 825 newtons to achieve an acceleration of 15 m/s². What does a force of 825 N look like in practical terms? Just imagine trying to push a heavy object such as a large box or a car. This calculation shows the kind of exertion needed during a sprinter's explosive start which is an essential part of their performance on the track.

Newton's Third Law

Newton's Third Law often summarized as "For every action, there is an equal and opposite reaction," is critical for understanding interactions between objects. In practical scenarios, we witness this law in action. When a sprinter pushes against the starting blocks, the blocks apply an equal and opposite force back onto the sprinter. This reactive force is what actually propels the sprinter forward. The comprehension of this law helps explain why sprinters use starting blocks. The blocks provide a stable platform to exert force against, maximizing the force that propels them forward. While it might seem that the sprinter propels themselves, it is actually the reaction force from the blocks that is responsible. By understanding Newton's Third Law, one can see that motion is not just about the action of the sprinter, but involves the interaction between the sprinter and the blocks, illustrating the integral role that external forces play in motion.