Science A-52 February 17th, 2001

Section 1: Basic physical concepts

Problem 1: Pile of coal example.

Problem 2: Tim Montgomery of the United States set a world record for the 100 m dash on September 14, 2002 completing the distance in a time of 9.78 s. Calculate his average speed in m/s and then convert the answer to miles/hour? Assuming that Montgomery accelerated to his average speed in the first 2 seconds, calculate his rate of acceleration and compare with gravity (Hint: assume the acceleration over the first two seconds is constant).

First we take the average speed as the change in distance with the change in time:

Now we convert the speed from meters per second to miles per hour. It helps to write out the units to make sure you haven’t forgotten terms:

We can calculate the acceleration simply if we assume constant acceleration over a time period:

Comparing this with gravity we find:

Problem: Assume that Montgomery increased his speed steadily over the course of his record-breaking run maintaining a constant acceleration. The speed when he broke the tape in this case would be equal to twice the average speed calculated in the previous problem. Suppose that he weighed 170 pounds. Calculate his kinetic energy at the end of the race.

First we need to get all the terms in the same units – we want mass in kilograms, speed in m/s.

Now we can plug terms into the equation for kinetic energy:

Note that the units are in Newton * meters or Joules. A Newton is a unit of force namely:

Also this is a good time to look back at the answer and think about it. Our assumption of constant acceleration gives Montgomery a final speed of:

!!

When’s the last time you say someone running 45 mph? – probably never so it’s clear that our assumption of constant acceleration is wrong.

Problem: A pole-vaulter seeks to generate as much speed as possible as he runs up to the bar. The aim is to convert the largest fraction of his kinetic energy as he can into bending the pole and then to use the energy stored in the pole to lift his body to do the work against gravity. Assume that the pole-vaulter has the same mass Montgomery. Assume that he is able to generate speed equal to the average of the speed achieved by Montgomery as calculated in the previous problem. Calculate the height to which the pole-vaulter is able to raise his center of mass if he succeeds in converting 100 % of his kinetic energy into work against gravity.

So for this problem we convert Montgomery’s average kinetic energy to potential energy:

From the physics primer we can find formulas for potential and kinetic energy. These are commonly used throughout the course so they would be useful to remember.

Manipulating the equation, we can express the max height of the pole-vaulter as a function of velocity and gravity.

Plugging in numbers we find:

In the notes we read that the highest jump was 20.14 feet by Sergey Bubka. We can infer that Sergey was either running faster than 10.2 m/s or that he did some work with his upper body during the jump.

Problem: The Chinese government if nearing completing of what will eventually be the world’s largest dam, the Three Gorges Dam on the Yangtze River. The dam will extend to a height of 181 m. Assume that 1m3 of water is allowed to overflow the dam and fall to the bottom on the other side.

(a) Calculate the speed of the water when it reaches the bottom.

(b) Calculate the kinetic energy of the water when it reaches the bottom

Again we want to equate the potential energy with kinetic energy. In this case the potential energy of the water at the top of the dam is equal to the kinetic energy of the water at the bottom of the dam:

Now manipulating the equation, we find this expression for velocity (Note: mass has been canceled out):

To find the Kinetic energy of the water we can either plug into the potential or kinetic energy formula since energy is assumed to be conserved (i.e. no air resistance):

Problem: The flow of water in the Yangtze River immediately upstream of the dam averages 60,000 m3/s. Of this, 20,000 m3/s, is used to drive turbines to generate electricity. The turbines are situated 125 m below the level of the water behind the dam. We refer to his as the hydraulic head for the water driving the turbines. Estimate the electrical power that would be realized if 100% of the potential energy of the water flowing through the turbines could be converted to electricity.

This is a similar problem to that above. First let’s find the kinetic energy of 1m3 of falling water then multiply by the flow rate:

Problem: Vector problem: An airplane flies SouthWest from Boston at 500mph for one hour. During the first half of the flight there was a 80mph wind blowing north-south, and during the second half of the flight there was a 120mph wind blowing from the northwest to the southeast. Calculate the coordinates of the plane after one hour. (Note: use x-direction as east, y-direction as north).

This problem is simplified by realizing that the wind is average, so we can just add each component. The flight track without wind gives us:

Now lets calculate the winds:

Now put it all together:

So the plane was blown significantly off course.

Problem: A barometer is used to measure pressure. How high would the water column be if it was filled with water?

Barometers work by balancing the forces of (1) weight of the fluid column above A with (2) the atmospheric pressure exerted all points of the surface (e.g. B). We can calculate how much water we’d need to balance the atmospheric pressure if we know the density of water:

As said above the force at A is the mass of the column of fluid times gravity. The force at B is the pressure of the atmosphere*area – in this case the area of the column at B

Breaking out the term on the left hand side we get:

Plugging this expression into the force balance we see that the area’s cancel:

Now we re-arrange terms to get an expression for height h:

Lastly we plug in numbers to get: