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 \(\mathrm{m} / \mathrm{s}^{2}\) . How much horizontal force must a \(55-\mathrm{kg}\) sprinter exert on the starting blocks during a start 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 a horizontal force of 825 N. The blocks propel the sprinter.

Step 1: Understand the Problem

We need to calculate the horizontal force exerted by the sprinter based on the given mass and acceleration using Newton's Second Law of Motion.

Step 2: Identify the Given Values

The acceleration of the sprinter is given as \(a = 15 \ \mathrm{m/s^2}\) and the mass of the sprinter is \(m = 55 \ \mathrm{kg}\). We are tasked to find the force \(F\).

Step 3: Use Newton's Second Law

Newton's Second Law states \( F = m \times a \), where \( F \) is the force, \( m \) is the mass, and \( a \) is the acceleration.

Step 4: Calculate the Force

Substitute the given values into the formula: \[ F = 55 \times 15 = 825 \ \mathrm{N} \]. Thus, the sprinter exerts a horizontal force of 825 Newtons on the starting blocks.

Step 5: Identify the Source of Propulsion

The force that propels the sprinter forward is due to the reaction force from the starting blocks according to Newton's Third Law. Therefore, the blocks exert an equal and opposite force to the sprinter.

Key Concepts

Force Calculation

When we talk about force calculation in the context of motion, we're usually considering Newton's Second Law of Motion. This law provides us a straightforward way to compute force when we know the values of mass and acceleration.

The formula is elegantly simple:

• \[ F = m \times a \]

where \( F \) represents force, \( m \) is mass, and \( a \) is acceleration.

In our sprinter scenario, we have an individual with a mass of 55 kg accelerating at a rate of 15 \( \mathrm{m/s^2} \). Plugging these numbers into our formula gives us:

• \[ F = 55 \times 15 \]
• Resulting in \[ F = 825 \ \mathrm{N} \]

Thus, the sprinter exerts 825 Newtons of force on the starting blocks to achieve this acceleration.

Acceleration

Acceleration, in the realm of physics, is the rate at which an object changes its velocity. It tells us how quickly an object is speeding up or slowing down. For instance, in our sprinter case, the acceleration is nearly horizontal and has a magnitude of 15 \( \mathrm{m/s^2} \).

Understanding acceleration is vital because:

• It allows us to calculate the force needed to produce such changes in motion.
• Acceleration tells us not just about speed but how rapidly that speed can change.

In many sports or real-world applications like the starting block scenario, swift acceleration is crucial as it defines how quickly an individual can reach competitive speeds.

Mass and Force Relationship

Mass and force share a relationship defined by Newton's Second Law, which connects mass to acceleration through force. Simply put:

• Greater mass requires a larger force to achieve the same acceleration.
• Larger force results in greater acceleration, assuming mass stays constant.

In terms of a sprinter, the mass of 55 kg plays a pivotal role. It tells us how much force is necessary to push the sprinter off the blocks. This is also where Newton's Third Law comes into play, emphasizing that for every action, there's an equal and opposite reaction.

The reaction force from the starting blocks is what propels the sprinter forward, demonstrating this perfect interaction between mass and force needed for acceleration. This relationship is not only a fundamental concept in physics but also critical in analyzing motion in everyday situations.