Find the components of vtot along the x- and y-axes in Figure 3.55.
Solution:
By isolating the vtot from the rest of the other vectors, we come up with the following figure. Also, the x and y-components are shown.
The resultant velocity has a magnitude of 6.72 m/s and is directed 49° from the positive x-axis. To solve for the x and y components, we just need to solve the legs of the right triangle formed by the three vectors. That is,
Find the magnitudes of velocities vA and vB in Figure 3.55
Solution:
Basically, we are given an oblique triangle. First, we shall determine the value of the interior angle at the intersection of vA and vB. We can solve this knowing that the sum of the interior angles of a triangle is 180°.
Show that the sum of the vectors discussed in Example 3.2 gives the result shown in Figure 3.24.
Solution:
So, we are given the two vectors shown below.
If we use the graphical method of adding vectors, we can join the two vectors using head-tail addition and come up with the following:
The resultant is drawn from the tail of the first vectors (the origin) to the head of the last vector. The resultant is shown in red in the figure below.
Show that the order of addition of three vectors does not affect their sum. Show this property by choosing any three vectors A, B, and C, all having different lengths and directions. Find the sum A + B + C then find their sum when added in a different order and show the result is the same. (There are five other orders in which A, B, and C can be added; choose only one.)
Solution:
Consider the three vectors shown in the figures below:
Vector A
Vector B
Vector C
First, we shall add them A+B+C. Using the head-tail or graphical method of vector addition, we have the figure shown below.
Now, let us try to find the sum of the three vectors by reordering vectors A, B, and C. Let us try to find the sum of C+B+A in that order. The result is shown below.
We can see that the resultant is the same directed from the origin upward. This proves that the resultant must be the same even if the vectors are added in different order.
Figure 2.68 shows the position graph for a particle for 6 s. (a) Draw the corresponding Velocity vs. Time graph. (b) What is the acceleration between 0 s and 2 s? (c) What happens to the acceleration at exactly 2 s?
Solution:
Part A
The velocity of the particle is the slope of the position vs time graph. Since the position graph is composed of straight lines, we can say that the velocity is constant for several time ranges.
Based on the data in the table, we can draw the velocity diagram
Part B
Since the velocity is constant between 0 seconds and 2 seconds, we say that the acceleration is 0.
Part C
Since there is a sudden change in velocity at exactly 2 seconds in a very short amount of time, we say that the acceleration is undefined in this case.
A graph of v(t) is shown for a world-class track sprinter in a 100-m race. (See Figure 2.67). (a) What is his average velocity for the first 4 s? (b) What is his instantaneous velocity at t=5 s? (c) What is his average acceleration between 0 and 4 s? (d) What is his time for the race?
Solution:
Part A
To find the average velocity over the straight line graph of the velocity vs time shown, we just need to locate the midpoint of the line. In this case, the average speed for the first 4 seconds is
So, the sprinter traveled a total of 24 meters in the first 4 seconds. He still needs to travel a distance of 76 meters to cover the total racing distance. At the constant rate of 12 m/s, he can run the remaining distance by
Estimate the average speed with which the hair on your head grows. Give your answer in both m/s and µm/hour. Briefly describe how you arrived at this estimate.
Estimate the average speed with which you go from home to campus via whatever mode of transportation you use most commonly. Give your answer in both mph and m/s. Briefly describe how you arrived at this estimate.
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