Question

If you ever slapped someone or got slapped yourself you probably remember the burning sensation. Imagine you had the unfortunate occasion of being slapped by an angry person, which caused the temperature of the affected area of your face to rise by \(2.4^{\circ} \mathrm{C}\) (ouch!). Assuming the slapping hand has a mass of \(0.9 \mathrm{kg}\) and about \(0.150 \mathrm{kg}\) of the tissue on the face and the hand is affected by the incident, estimate the velocity of the hand just before impact. Take the specific heat of the tissue to be \(3.8 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\).

Short answer

Question: Estimate the velocity of the slapping hand just before impact. Answer: The estimated velocity of the hand just before impact was about 55.1 m/s.

Step-by-step solution

Recall the equation for heat transfer

We can start by using the heat transfer equation, which states that the heat transferred \(Q\) is equal to the product of the mass \(m\), specific heat \(c\), and the change in temperature \(\Delta T\). It looks like this: \(Q = mc\Delta T\)

Calculate the heat transferred

Now let's calculate the heat transferred in this case, using the values given in the exercise. We have the mass of the affected tissue \(m_t = 0.150 \mathrm{kg}\), the specific heat \(c = 3.8 \mathrm{kJ / kg \cdot K}\), and the temperature increase \(\Delta T = 2.4^{\circ}\mathrm{C}\). \(Q = m_t c \Delta T = 0.150 \mathrm{kg} \cdot 3.8 \mathrm{kJ / kg \cdot K} \cdot 2.4^{\circ}\mathrm{C} = 1.368 \mathrm{kJ}\)

Convert the heat transferred into kinetic energy

We can assume that the heat transferred to the face comes from the kinetic energy of the hand, so we have: \(Q = \dfrac{1}{2}m_h v^2\) Where \(m_h\) is the mass of the hand, and \(v\) is the velocity just before impact. In our case, \(m_h = 0.9 \mathrm{kg}\). To make the units consistent, we need to convert \(Q\) from kJ to J: \(Q = 1.368 \mathrm{kJ} = 1368 \mathrm{J}\)

Solve for the velocity of the hand

Now we can solve for the velocity \(v\) of the hand just before impact: \(1368 \mathrm{J} = \dfrac{1}{2}(0.9 \mathrm{kg}) v^2\) To find \(v\), we can divide both sides by \(0.45 \mathrm{kg}\), then take the square root: \(v^2 = 1368 \mathrm{J} / 0.45 \mathrm{kg} \approx 3040\) \(v = \sqrt{3040} \approx 55.1 \mathrm{m/s}\) So, the estimated velocity of the hand just before impact was about \(55.1 \mathrm{m/s}\).

Key concepts

Specific Heat Capacity

The specific heat capacity is a property of a material that indicates how much heat is needed to raise the temperature of one kilogram of the material by one degree Celsius (or Kelvin since the temperature increment is the same). It's symbolized by the letter 'c' and typically expressed in units of joules per kilogram per degree Celsius (J/kg°C) or kilojoules per kilogram per Kelvin (kJ/kg·K).

The larger the specific heat capacity value, the more heat is required to change the temperature, making it a crucial factor in understanding heat transfer in thermodynamics. For example, water has a high specific heat capacity, which is why it's effective at absorbing heat without experiencing a rapid temperature increase, a trait essential in many industrial processes and climate regulation.

Specific heat capacity plays a pivotal role in heat exchange scenarios. When it comes to our slapping example, the specific heat capacity of the facial tissue is used to calculate the amount of heat transferred during the slap, which in turn, provides insight into the kinetics of the event.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It's a form of energy that every moving object has, from a giant rolling boulder to a person's hand just before a slap. The formula for kinetic energy (\( KE \text{ or } Q \text{ in a collision scenario} \) is given by \frac{1}{2}mv^2\text{. Here }m\text{ is the mass of the object, and }v\text{ is the velocity of the object.}

When you apply this concept to calculate the velocity of a moving hand, you're essentially reversing the common use of the formula: you find the velocity by knowing the energy amount transferred during the impact. The kinetic energy can be converted into other forms of energy upon impact – like heat, in the case of the slapping hand – which often happens during collisions or interactions between objects. In the example, the energy transfer provides a very practical measure of the hand's velocity, underlining the hand's capacity for doing work (in this case, the unfortunate work of causing facial tissue temperature to rise).

Temperature Change

Temperature change is indicative of a transfer of thermal energy, which might occur due to various processes, such as conduction, convection, or radiation. Whenever an object or substance undergoes a change in temperature, it's a sign that heat, denoted as 'Q' in thermodynamics, has either been absorbed or released. The temperature change can be denoted as △T in equations.

The relationship between heat, mass, specific heat capacity, and temperature change is neatly summarized by the equation Q = mc△T, where 'm' is the mass of the substance, and 'c' is the specific heat capacity. This equation highlights how heat transfer is dependent on both the material properties (specific heat capacity) and the mass involved in the process.

Going back to our example, when the slap causes a temperature rise in the facial tissue, it's reflective of the heat – the energy transferred from the kinetic energy of the hand. By calculating how much the temperature has changed, we can work backwards to deduce the quantity of energy transferred and, subsequently, the kinetic energy of the hand pre-impact. This understanding is fundamental in fields extending from basic physics demonstrations to engineering applications where temperature regulation and energy conversion are crucial.