Problem 64 - Physics for Scientists and Engineers with Modern Physics

Question

(I) An 85 -kg football player traveling 5.0 \(\mathrm{m} / \mathrm{s}\) is stopped in 1.0 \(\mathrm{s}\) by a tackler. (a) What is the original kinetic energy of the player? \((b)\) What average power is required to stop him?

Step-by-Step Solution

Answer

(a) 1062.5 J; (b) 1062.5 W

Step 1: Calculate Kinetic Energy

The original kinetic energy \( KE \) of the football player can be calculated using the formula: \( KE = \frac{1}{2}mv^2 \). Here, \( m = 85 \text{ kg} \) (mass) and \( v = 5.0 \text{ m/s} \) (velocity). Substitute the values: \( KE = \frac{1}{2} \times 85 \times (5.0)^2 \). Perform the calculation: \( KE = 0.5 \times 85 \times 25 = 1062.5 \text{ J} \).

Step 2: Find Work Done to Stop the Player

The work done to stop the player is equal to the original kinetic energy, since all the kinetic energy is dissipated as work. Thus, \( W = 1062.5 \text{ J} \).

Step 3: Calculate Average Power

Average power \( P \) is calculated by dividing the work done \( W \) by the time \( t \) it takes to stop the player. The formula is \( P = \frac{W}{t} \). Given the work done is \( 1062.5 \text{ J} \) and the time taken is \( 1.0 \text{ s} \), substitute these values: \( P = \frac{1062.5}{1.0} = 1062.5 \text{ W} \).

Key Concepts

Kinetic Energy

Kinetic energy is a concept from mechanics that describes the energy an object possesses due to its motion. Imagine a football player darting across the field. The faster he moves or the heavier he is, the more kinetic energy he has. The kinetic energy (KE) of an object can be determined through the formula:\[KE = \frac{1}{2}mv^2\]where:

• \(m\) is the mass of the object in kilograms (kg).
• \(v\) is the velocity or speed of the object in meters per second (m/s).

In this example, the football player weighs 85 kg and is running at a speed of 5.0 m/s. By plugging these numbers into the formula, we calculated his kinetic energy as 1062.5 Joules (J). This energy is what needs to be overcome to bring him to a stop.

Average Power

Average power is another important concept in mechanics that deals with how quickly work is done or energy is transferred. In simpler terms, it's about the rate of doing work. The formula used to calculate average power (P) is:\[P = \frac{W}{t}\]where:

• \(W\) is the work done in Joules (J).
• \(t\) is the time taken in seconds (s).

In the context of the football exercise, we know that all the player’s kinetic energy (1062.5 J) is used to stop him, and the time taken to stop is 1.0 second. Thus, the average power required to stop the player is 1062.5 Watts (W). This aligns with the understanding that the quicker the energy is spent, the greater the power required.

Work-Energy Theorem

The work-energy theorem is a principle in mechanics that explains the relationship between work done and the energy change in a system. According to this theorem, the work done on an object is equal to the change in its kinetic energy. The formula is:\[W = \Delta KE = KE_{final} - KE_{initial}\]For an object coming to a stop, like our football player, the initial kinetic energy is converted into work done to stop the player, with the final kinetic energy becoming zero.In this scenario:

• Work done (W) is 1062.5 J, which is equal to the initial kinetic energy.
• The change in kinetic energy is also 1062.5 J, as the player's velocity decreases from 5.0 m/s to 0 m/s.

This theorem beautifully ties together the concept of work and how it results in changes to kinetic energy, providing a deeper understanding of energy transfer in mechanical processes.