Problem 21 - University Physics with Modern Physics

Question

World-class sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has magnitude 15 m/s\(^2\). How much horizontal force must a 55-kg sprinter exert on the starting blocks to produce this acceleration? Which body exerts the force that propels the sprinter: the blocks or the sprinter herself?

Step-by-Step Solution

Answer

The sprinter exerts an 825 N force on the blocks to propel herself forward.

Step 1: Identify the Given Values

First, note down the known values from the problem. The acceleration of the sprinter, \(a\), is given as 15 m/s\(^2\), and the mass of the sprinter, \(m\), is given as 55 kg.

Step 2: Recall Newton's Second Law of Motion

Newton's Second Law states \( F = ma \), where \( F \) is the force, \( m \) is the mass, and \( a \) is the acceleration. This formula will allow us to calculate the horizontal force exerted.

Step 3: Calculate the Horizontal Force

Substitute the given values into the formula: \( F = 55 \, \text{kg} \times 15 \, \text{m/s}^2 \). Calculate the result to find the force.

Step 4: Perform the Calculation

Compute the multiplication: \( F = 55 \, \text{kg} \times 15 \, \text{m/s}^2 = 825 \, \text{N} \). So, the horizontal force exerted is 825 Newtons.

Step 5: Identify Which Body Exerts the Force

According to Newton's Third Law, for every action, there is an equal and opposite reaction. Therefore, the sprinter exerts the force on the block, and the block exerts an equal force that propels the sprinter forward.

Key Concepts

Horizontal Force Calculation

Understanding how to calculate horizontal force is crucial when analyzing dynamics involving moving objects. By using Newton's Second Law of Motion, which is a fundamental principle in physics, we can precisely determine the force exerted in these scenarios.
Newton's Second Law is formulated as \( F = ma \). This tells us that the force \( F \) acting on an object is the product of its mass \( m \) and its acceleration \( a \). For the sprinter in our example:

• Mass \( m = 55 \) kg
• Acceleration \( a = 15 \) m/s\(^2\)

Simplifying this multiplication, \( F = 55 \times 15 = 825 \) Newtons, reveals the horizontal force needed for the sprinter to achieve such a rapid start. Understanding this relationship helps in predicting how objects will move under various forces.

Acceleration and Mass

Acceleration and mass are directly related to force through Newton's Second Law of Motion. The mass of an object tells us how much matter it contains, while acceleration indicates how quickly its velocity is changing. Together, they determine how much force is required to change the object's motion.
When thinking about acceleration, it's useful to remember:

• Acceleration depends on the amount of force applied. More force means more acceleration.
• Heavier objects (greater mass) require more force to achieve the same acceleration as lighter ones.

In the case of our 55-kg sprinter, achieving an acceleration of 15 m/s\(^2\) requires a calculated force of 825 N. This result shows the role both mass and acceleration play in motion dynamics.
It explains why a small object can accelerate faster with less force, while a larger object needs more force to achieve the same rate of change in its velocity.

Reaction Forces

Reaction forces are rooted in Newton's Third Law of Motion, which states that for every action, there is an equal and opposite reaction. When a sprinter pushes against the starting blocks, the blocks exert an equal force back, propelling the sprinter forward.
This interaction can be understood as:

• The sprinter applies a force on the blocks in the backward direction.
• The blocks exert an equal and opposite force forward, which moves the sprinter off the block.

This concept highlights that while it seems the sprinter's legs are solely responsible for movement, it's actually this exchange of forces that enables her progress.
Understanding reaction forces uncovers how interactions not just initiate motion, but also balance it in such a way that complex motions are achievable with coordinated actions between different bodies.