Question

a) A is a quantity with two significant figures. B has 4 significant figures.
If  C = A + B, D = A*B, and E=B/A
What should the number of significant figures of C, D and E be?

b) Compare and contrast the following pairs of definitions, one set at a time. For each set of definitions, describe what kind of measurements ( time, positions etc) must you do in order to compute the quantity based on their definitions. Be as specific as you can. All resulting quantities must be expressed in certain number of significant figures. Why is that?

1. Average speed vs average velocity
2. Instantaneous speed vs instantaneous velocity
3. Average acceleration vs Instantaneous acceleration

Answer 1

a)

A- 2 significant figures

B- 4 significant figures

C = A + B

For addition, the result should have decimal places equaling the minimum number of decimal places among the two numbers being added. So, the number of significant digits should be equal to the all the digits before the decimal and only those digits after the decimal which equal to the minimum number of decimals.

D = A*B

A has minimum significant figures which is 2, and this minimum should be the number of significant figures for multiplication. So, there should be 2 significant figures for D.

E = B/A

A has minimum significant figures which is 2, and this minimum should be the number of significant figures for division. So, there should be 2 significant figures for D.

b)

Average speed vs Average Velocity

Average speed = Total Distance/ Total time taken

So, to find average speed, we need to find the ratio between the total distance covered by the total time taken for that.

Average Velocity = Total Displacement/ Total Time taken

So, to find average velocity we need the ratio between the total displacement, i.e. the separation between the initial and final points. and the total time taken.

Instantaneous speed vs Instantaneous Velocity

Instantaneous speed = Instantaneous distance/ Time

Instantaneous speed is the speed at a particular instant, so to get that we require distance at the instant and find the ratio with time.

Instantaneous velocity = Instantaneous displacement/ Time

Instantaneous velocity is the velocity at a particular instant, so to get that we require displacement at the instant and find the ratio with time.

Average acceleration vs Instantaneous acceleration

Average acceleration = (Final Velocity - Initial Velocity)/ time taken

Average acceleration is the time taken for the velocity to reach from its initial to final value, so we can get it by taking the ratio of the difference between the velocities and the time taken

Instantaneous acceleration = (Velocity at the instant - initial Velocity)/ time taken

Instantaneous acceleration is the acceleration at a given instant. So, to obtain it we require find the difference between the velocity at that instant and the initial velocity and the time taken for that.

Answer 2

a)

For addition and subtraction, the number of decimal places in the result should be the same as the number of decimal places in the quantity with the fewest decimal places. Since A has two significant figures, C should have two significant figures.

For multiplication and division, the number of significant figures in the result should be the same as the quantity with the fewest significant figures. Since A has two significant figures and B has four significant figures, D should have two significant figures and E should have two significant figures.

b)

Average speed is defined as the total distance traveled divided by the total time taken. In order to compute average speed, measurements of distance and time must be taken. The resulting quantity should have the same number of significant figures as the least precise measurement.

Average velocity is defined as the displacement (change in position) divided by the total time taken. In order to compute average velocity, measurements of displacement and time must be taken. The resulting quantity should have the same number of significant figures as the least precise measurement.

Instantaneous speed is the speed of an object at a specific instant in time. In order to compute instantaneous speed, the distance traveled by the object over a very short period of time must be measured. The resulting quantity should have the same number of significant figures as the least precise measurement.

Instantaneous velocity is the velocity of an object at a specific instant in time. In order to compute instantaneous velocity, the displacement of the object over a very short period of time must be measured. The resulting quantity should have the same number of significant figures as the least precise measurement.

Average acceleration is defined as the change in velocity divided by the total time taken. In order to compute average acceleration, measurements of velocity and time must be taken. The resulting quantity should have the same number of significant figures as the least precise measurement.

Instantaneous acceleration is the acceleration of an object at a specific instant in time. In order to compute instantaneous acceleration, the change in velocity over a very short period of time must be measured. The resulting quantity should have the same number of significant figures as the least precise measurement.