If cold gas is added to hot gas in a rigid sealed container, what happens to the pressure? Does it increase, decrease, remain constant, or was there insufficient information to answer the question? Assume both gases are ideal.
The pressure will increase, regardless of the temperature.
If two trains leave a station simultaneously on tracks at a 60 degree angle to each other, how fast will they be moving apart in 2 hours if they are travelling at 80 km/h, and 100 km/h, respectively?
The easiest way to approach this question is to recognize that a triangle is formed by the train station and each of the trains. Lets call the distance the first train has traveled A, the distance the second train has traveled B, and the distance between them C. Cosine law relates these distances by the equation C²=A²+B²-(2)(A)(B)cos(60). Because the cosine of 60 degrees is 1/2, it cancels with the 2 to result in C²=A²+B²-AB. To solve the problem, we must find dC/dt by implicitly differentiating with respect to t. This results in: (2)(C)(dC/dt)=(2)(A)(dA/dt)+(2)(B)(dB/dt)-(A)(dB/dt)-(B)(dA/dt) At this point, we can substitute in numbers. dA/dt is 80km/h, dB/dt is 100km/h, A is 80km/h*2h=160km, B is 100km/h*2h=200km, and C is sqrt(A²+B²-AB)=sqrt(33600) km. This gives us: 2sqrt(33600)(dC/dt)=(2)(160)(80)+(2)(200)(100)-(160)(100)-(200)(80) 2sqrt(33600)(dC/dt)=33600 dC/dt=33600/(2sqrt(33600)) dC/dt=sqrt(8400)=92 km/h
A cylinder and a block are each at the top of a ramp. At the same time, the cylinder starts rolling and the block starts sliding down, neglecting friction. Which will reach the bottom first, and why?
The block will reach the bottom first. The easiest way one could approach it is by looking at how the potential energy is broken up into its various components: linear and rotational. The block will convert its entire potential energy into linear kinetic energy, which means that it is all being used to make it reach the bottom as fast as possible. However, the cylinder converts some of its potential energy into rotational kinetic energy, which does not help it move down the ramp. Because some of its energy is wasted, it must be slower than the block.