~23 students responded to JiTT 1 and ~24 students responded to Post-Lab 1
In the United States, according to data provided by the CDC, the average person's lifespan increased from 77.7 years in 2006 to 77.9 years in 2007. More recently the CIA has calculated its estimate for the average person's lifespan in the United States to be 78.11 years. In trying to find the how many heart beats occur in an average person's lifespan, one needs to know this information, as well as, the time duration between beats. Table 1.3 in our textbook lists this value as 8x10^-1. Thus, to calculate how how many beats occur one must convert units, where all ratios in parentheses are equal to 1. [78.11 yrs (365.2 days/yr) (24 hrs/day) (60 minutes/hr) (60 s/minute)] x (1 heart beat/8x10^-1) = 3 x 10^9 2.
Our textbook reads, "Dimensions denote the physical nature of a quantity." Dimensional analysis is a procedure used by physicists "to check a specific equation to see if it matches expectations." Because dimensions can be substituted into mathematical equations as algebraic quantities, physicists use dimensional analysis to make sure that terms added and subtracted have the same units. Dimensional analysis also checks that equivalence statements are only made between terms with the same units. Physicists perform dimensional analysis either by substituting the dimensions directly into an equation and canceling, or by creating a expression with a proportionality constant, substituting the dimensions, and solving for unknowns.
sin(x) is equivalent to x within two significant digits when -.32<x<.32 radians. I calculated this by comparing the graphs of sin (x) and x on my calculator, and then checking the answer by plugging in values both higher and lower. tan(x) is equivalent to x within two significant digits when -.24<x<.24 radians. I calculated this by comparing the graphs of tan (x) and x on my calculator, and then checking the answer by plugging in values both higher and lower. tan(x) is equivalent to sin(x) within two significant digits when -.24 + or - n(2pi) < x< .24 + or - n(2 pi). I calculated this answer by comparing the graphs of sin(x) and tan(x) on my calculator. I confirmed the answer by plugging in values both higher and lower.
There are two points at which velocity and acceleration are in the same direction. One is when the ball is falling back to the ground after reaching its peak. As velocity is defined as the change in position divided by the change in time, it could be represented by a downward arrow. Acceleration is defined as the change is velocity over the change in time. Because the object is falling, there must be a gravitation force pulling it back to the ground. Gravitational acceleration can be represented by a downward arrow. The second is shortly after the ball hits the ground. There is intense upward acceleration. This is eventually overcome by the downward gravitational acceleration. The velocity and acceleration are in opposite directions as the ball slows down after bouncing back off the ground. Here, the ball continues in the upward direction and can thus be represented by an upward arrow. However, because the ball is climbing upward and slowing down, the acceleration will be negative and can be represented as a downward arrow. The velocity is zero at two specific time points. The ball's instantaneous velocity is zero at the peak of it's climb and at the moment the ball hits the ground. At this moments, the ball's position is not changing and thus the velocity is zero. The acceleration is briefly zero. Because the acceleration becomes intensely positive after every bounce and then returns negative, the acceleration must be zero at some points.
I estimated that the average number of heartbeats per minute is 100 and that the average person lives about 70 years. I then used dimensional analysis to find that the average person's heart beats 3.7 x 10^9 times in their lifetime.
Dimensional analysis refers to the process by which scientists can determine what units of measurement need to accompany a numerical value.
I graphed both functions on a calculator and observed where the ranges of the functions began to differ. Sin(x)~=x to two significant digits for any angle smaller than 18 degrees. Tan(x)~=x for any angle smaller than 14 degrees. Tan(x)~=sin(x) for any angle smaller than 14 degrees.
Velocity and acceleration are not always in the same direction nor are they always in the opposite direction. When the ball is traveling upwards, the velocity and acceleration are in different directions. The acceleration is downwards while velocity is upwards. When the ball is traveling downwards, both acceleration and velocity are downwards. The velocity is zero when the ball changes from traveling in an upward direction to traveling in a downward direction. The acceleration is never zero because the velocity of the ball changes at a constant rate.
According to google, the average human heart beats at about 70 bpm and the average life expectancy in the United States is about 78 years. If we multiply 70 by 1440, the number of minutes in one day, we get 100800 heartbeats per day for the average U.S. citizen. Multiplying this number by 365 yields 36,792,000, the number of heartbeats per year. Finally, multiplying this number by 78 gives us 2,869,776,000 (or 2.86x10^9), the amount of heartbeats that occur over the average American life.
“Dimensional analysis” is a method for determining if both sides of an expression have the same dimension.
Velocity and acceleration are not always in the same direction. When the ball is bouncing upwards, but the force of gravity is pulling it downwards and causing it to slow down, or have a negative acceleration. Velocity and acceleration are not always in opposite directions; when the ball is falling downwards, they are both in the same direction. The velocity of the ball is zero at the apex of its bounce and also when at the moment it contacts the ground. The acceleration of the ball is never zero, because the velocity is constantly changing.
Student A: Dimensional analysis, while easy to grasp conceptually, was very difficult for me to articulate and might be worth going over once in class so that we have a formal definition to return to.
Student B: Question 3 was very confusing. I was not sure what the question was asking for and if the symbol ~ had any significance. I was not sure if the symbol referred to order of magnitude as discussed in the reading or if it meant to approximate my answer.
Student C: I had a very hard time understanding question 3. Could we please go over this in class?