Week 3 Spped And Velocity

Question 1

Position vector

Answer 1

In mechanics, position is usually defined via the “position vector” . This is the vector that has its tail in the referent point and its head in the point being tracked. Note that:

• The components of the position vector are the coordinates of the point being tracked.

•The magnitude of the position vector is the distance between the referent point and the point being tracked .

Question 2

Coordinates and position vector in movement

Answer 2

the coordinates of a moving point change with time (t) They are said to be functions of time. Mathematically this is expressed as

X(t), y(t) z(t)
Since the coordinates are components of the postion vector, this means that the position vector is also a function of Time, which is expressed as
R (t)

Question 3

Trajectory

Answer 3

The collection of successive positions of a moving point is called a trajectory
It can also be said that the trajectory is a line drawn by the head of the position vector as the point moves

Question 4

Average speed

Answer 4

Average speed over a certain interval of time (^t)is defined as the distance travelled (^s) divided by the time interval
It is a scalar quantity with the SI unit of m/s

Average speed gives general information on the rate at which a moving object covers the distance over time. It does not tel us anything about the direction of movement

V= s/ t2-t1

Note that no matter how small the time interval of measurement is, there will
always be an error in estimating instantaneous speed! The error will diminish as
the size of time interval is reduced, but it will never vanish completely.

V = s (t+^t) - s (t- ^t). / 2^t

Question 5

Velocity

Answer 5

Velocity is the rate at which the position (vector) changes with time. Mathematically, this can be expressed as:
V= d — ^ x
^t —^t

Unlike the speed velocity is a vector and it tells us both how fast an object is moving at any instant of time and in which direction it is moving

Question 6

Velocity vector in component form

Answer 6

Components of the velocity vector are the rates at which the three coordinates change in time , i.e., the slopes of the x versus time, y versus time , and z versus time curves.

The magnitude of the velocity vector is the instantaneous speed.

The direction of the velocity vector is always along the tangent to the trajectory.

Question 7

Position from velocity data

Answer 7

Position can be calculated from velocity data by using the mathematical operation called integration.
For each coordinate this is expressed mathematically as:
X (t2)= x(t1) + Vxt

In practical terms, the charge of coordinate over a certain period of time is equal to the area beneath the curve of the corresponding velocity component plotted against time.

Question 8

How can we measure speed/velocity

Answer 8

Timing gates
Video footage
Radar
Linear transducers

Question 9

Key points

Answer 9

Position of a point in space is defined by three independent coordinates.

“Position vector” is the vector whose components are the coordinates of the point being tracked.

Vector algebra is useful for extracting information from the experimentally obtained position data.

Path length

Displacement

Speed is the rate at which a moving object covers distance over time.

Velocity is the rate at which an object changes position over time.

Resultant of the velocity vector is the (instantaneous) speeds.

The speed and direction of motion can be calculated from experimentally obtained position data (coordinates).