Vectors

Force, velocity, momentum, and acceleration, have both size and direction. These quantities are vectors. The addition of vectors must be understood to solve problems with two or more forces acting at a point, a plane landing in a cross wind, a boy swimming a river, or a mass in circular motion.


The sum of two vectors

Vectors are added using diagrams or by adding the numbers in a column vector symbol. To practice the addition of vectors with an a Applet click here.


Force is a vector

When three or more forces are balanced (in equilibrium) at a point, the vector sum of the forces is zero. For physics types, the idea is trivial, but it makes a nice computer drawing exercise.

Three identical springs are stretched on a circular hoop and photographed. A fourth identical spring is shown as at zero extension. The unstretched length of each spring is marked. Force vectors are drawn to scale [length, and direction] as the extensions of the springs. The three force vectors are moved to form the expected closed triangle.When four springs are connected to a single point, the four force vectors add to zero.

Try it for yourself ...

1 Copy an image to AppleWorks.

2 Draw lines to represent the unextended spring length and the extensions.

3 Use the free rotate function to turn the lines and make up the vector diagram.

4 Try it with four vectors.


Velocity is a vector

Displacement, velocity, momentum and acceleration are all vectors. The following examples must be in every IB candidates exam tool box.

Examples:

1 Swimming a river

A young man; handsome lad, but short of a ferry ticket, is stuck on the bank of a river in a far off country. The love of his life is on the opposite side (in Laos) with the lunch.

The river is 100 meters wide and flowing to the right (as he sees it) at 3 m/s. He has new flippers and can swim at 5 m/s so he fancies his chances. Give advice. In what direction must he swim to pass straight across the river and arrive on the opposite bank. How long will that take?

For answers click next

2 Landing a plane

Landing a plane in a cross wind is tricky. The plane must be flown in the air at an angle to the runway so that it follows the right line on the ground. In the clip shown the pilot is landing on a North-South runway with an air-speed of 80 m/s. He is flying in a strong westerly cross wind of 20 m/s (at right angles to the runway).

In what direction must he fly, and what is his ground speed on landing.

Answer


The difference of two vectors

The difference in any quantity is the final value minus the initial value.

Do it with money: 500 Baht this week - 300 Baht next week - amounts to a pay cut of 200 Baht.

300 - 500 = - 200

Momentum is a vector

Examples:

[This one is a favorite exam question.]

1 A ball of mass m strikes a vertical wall at normal incidence and speed v. It rebounds without loss of energy. The collision is said to be elastic.

a What is the momentum change of the ball and what is the impulse applied to the wall?

Answer

2 The same ball of mass m strikes the frictionless wall with speed v at an angle. It rebounds without loss of kinetic energy.

b What is the momentum change of the ball and what is the the impulse applied to the wall?

Answer

Note: imagine you are the wall ... you turn round and close your eyes, The ball hits you on the back of the head. There is no friction between your head and the ball. The impulse is perpendicular to the back of your head independent of the angle of incidence. You cannot even begin to guess who threw the ball.


Acceleration is a vector

Example

A mass m is moving in a circle at constant speed v.

a Show from first principles that the acceleration is towards the center.

b Show that the magnitude (size) of the acceleration is given by....

a = v2/r

c Show that the centripetal force must be given by.f = mv2/r towards the center/

Answers


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