How many moles of air are there in a room of dimensions 3 m x 5 m x 4 m? You may assume that the air inside of the room is at 1 atm of pressure and is at a temperature of 20 degrees Celsius.
To solve this problem, we can employ the Ideal Gas Law. It says that if an ideal gas is kept within a closed container, then PV = nRT, where P is the pressure of the gas in Pa, V is the volume of the gas in m^3, n is the number of moles of the gas, R = 8.314 J/(mol*K), and T is the temperature of the gas in K. 1 atm = 1.013 x 10^5 Pa, and so we'll need to let P = 1.013 x 10^5 Pa for this problem. 3 m x 5 m x 4 m = 60 m^3, and so we'll need to let V = 60 m^3 for this problem. To convert a temperature from degrees Celsius to K, we need to add 273. So a temperature of 20 degrees Celsius is equivalent to a temperature of 20 + 273 = 293 K. Therefore, we need to let T = 293 K for this problem. Plugging all of the values into the Ideal Gas Law, we get (1.013 x 10^5 Pa)(60 m^3) = n(8.314 J/(mol*K))(293 K) . We can solve this equation for n by dividing both sides of the equation by (8.314 J/(mol*K))(293 K). Doing so yields n = 2495 mol . Therefore, contained within that modestly sized room, there are a whopping 2495 moles of air!
Peyton Manning runs at a speed of 4 m/s along a football field and ultimately crashes into Tom Brady, who is standing on the field. Peyton Manning has a mass of 104 kg. Tom Brady has a mass of 102 kg. If Peyton Manning and Tom Brady stick together after the collision, then what is their speed after the collision?
We can solve this problem by applying the Law of Conservation of Momentum. It states that the total momentum of a system remains the same unless a net outside force acts on the system. Here, during the brief collision between the two athletes, we'd expect that any unbalanced outside forces will have a negligible effect on the situation. Therefore, their total momentum will remain the same. The momentum p of an object of mass m and velocity v is given by the equation p = mv. Therefore, the momentum of Peyton Manning before the collision is (104 kg)(4 m/s) = 416 kg*m/s. Because Tom Brady is at rest before the collision, his velocity is 0, and his momentum is (102 kg)(0) = 0. Thus, the total momentum of the two athletes before the collision is just 416 kg*m/s. After the collision, both athletes are moving at the same velocity. So it's like they are a single object moving along with some velocity v. The total mass of this two-athlete "blob" is 104 kg + 102 kg = 206 kg. Hence the total momentum of the two athletes after the collision is (206 kg)v. Finally, we can set the total momentum before the collision equal to the total momentum after the collision, and we have the equation 416 kg*m/s = (206 kg)v . To solve this equation for v, we can divide both sides by 206 kg. This yields 2.02 m/s = v . So after this collision, the two athletes run off at a speed of 2.02 m/s.
On a hot summer day, why is it more comfortable to wear a white shirt than it is to wear a black shirt?
A white shirt appears white to us because it reflects all colors of light. On the other hand, a black shirt absorbs all colors of light. The light incident on these shirts from the Sun is thermal radiation; it is the means through which the Sun transfers heat to the shirts. Therefore, the black shirt heats up more quickly than the white shirt does, as it is absorbing more radiation from the Sun. On a hot summer day, though, we want to keep cool. Hence it is more comfortable for us to wear the white shirt than it is to wear the black shirt on the hot summer day.