Show that the sum of the vectors discussed in the example Subtracting Vectors Graphically: A Woman Sailing a Boat 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.

The result is in conformity with that in figure 3.24 shown on the question shown above.

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 figure below:

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 compose of straight lines, we can say that the velocity is constant for several time ranges.

Time Ranges

Slope of the position Graph/Velocity

0 to 2 seconds

2 to 3 seconds

3 to 5 seconds

5 to 5 seconds

Based from 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 for 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

Part B

Looking at the graph, the velocity at exactly 5 seconds is 12 m/s.

Part C

If we are given the velocity-time graph, we can solve for the acceleration by solving for the slope of the line.

Consider the line from time zero to time, t=4 seconds. The slope, or acceleration, is

Part D

For the first 4 seconds, the distance traveled is equal to the area under the curve.

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

(a) What is the maximum frictional force in the knee joint of a person who supports 66.0 kg of her mass on that knee?

(b) During strenuous exercise, it is possible to exert forces to the joints that are easily ten times greater than the weight being supported. What is the maximum force of friction under such conditions? The frictional forces in joints are relatively small in all circumstances except when the joints deteriorate, such as from injury or arthritis. Increased frictional forces can cause further damage and pain.

(a) When rebuilding her car’s engine, a physics major must exert 300 N of force to insert a dry steel piston into a steel cylinder. What is the normal force between the piston and cylinder?

(b) What force would she have to exert if the steel parts were oiled?

A physics major is cooking breakfast when he notices that the frictional force between his steel spatula and his Teflon frying pan is only 0.200 N. Knowing the coefficient of kinetic friction between the two materials, he quickly calculates the normal force. What is it?

A powerful motorcycle can produce an acceleration of 3.50m/s² while traveling at 90.0 km/h. At that speed the forces resisting motion, including friction and air resistance, total 400 N. (Air resistance is analogous to air friction. It always opposes the motion of an object.) What force does the motorcycle exert backward on the ground to produce its acceleration if the mass of the motorcycle with the rider is 245 kg?

Suppose two children push horizontally, but in exactly opposite directions, on a third child in a wagon. The first child exerts a force of 75.0 N, the second a force of 90.0 N, friction is 12.0 N, and the mass of the third child plus wagon is 23.0 kg.

(a) What is the system of interest if the acceleration of the child in the wagon is to be calculated?

(b) Draw a free-body diagram, including all forces acting on the system.

(c) Calculate the acceleration.

(d) What would the acceleration be if friction were 15.0 N?

What deceleration of the rocket sled if it comes to rest in 1.1 s from a speed of 1000 km/h? (Such deceleration caused one test subject to black out and have temporary blindness.)