KINEMATICS information

1. x(t) is called displacement and it denotes the position of a body at time. If the
displacement is positive then that body is to the right of the chosen origin and if
negative, then it is to the left.
2. If a body is moving with average speed v then in time t it will cover a distance d=vt.
But, in fact, the speed of a car changes from time to time and so one should limit the
use of this formula to small time differences only. So, more accurately, one defines an
average speed over the small time interval Δt as:
average speed distance travelled in time t
t
Δ
=
Δ
3. We define instantaneous velocity at any time t as:
2 1
2 1
v x(t ) x(t ) x
t t t
− Δ
= ≡
− Δ
.
Here and are both very small quantities that tend to zero but their
ratio v does not.
Δx Δt
4. Just as we have defined velocity as the rate of change of distance, similarly we can
define instantaneous acceleration at any time t as:
2 1
2 1
a v(t) v(t) v
t t t
− Δ
= ≡
− Δ
.
Here v and are both very small quantities that tend to zero but their
ratio is not zero, in general. Negative acceleration is called deceleration.
The speed of a decelerating body decreases with
t
a
Δ Δ
time.
5. Some students are puzzled by the fact that a body can have a very large acceleration
but can be standing still at a given time. In fact, it can be moving in the opposite
direction to its acceleration. There is actually nothing strange here because position,
velocity, and acceleration are independent quantities. This means that specifying one
does not specify the other.
13. Two vectors can be added together geometrically. We take any one vector, move
it without changing its direction so that both vectors start from the same point, and
then make a parallelogram. The diagonal of the parallelogram is the resultant.
6. For constant speed and a body that is at x=0 at time t=0, x increases linearly with
time, x∝t (or x=vt). 0 If the body is at position x at time t= 0, then at
0 time t, x =x +vt.
7. For constant acceleration and a body that starts from rest at t = 0, v increases
0
0
linearly with time, v (or v ). If the body has speed v at 0, then at
time , v v .
t at t
t at
∝ = =
= +
8. We know in (6) above how far a body moving at constant speed moves in time t. But
what if the body is changing its speed? If the speed is increasing linearly (i.e. constant
acceleration), then the answer is particularly simple: just use the same formula as in
(6) but use the average speed: 0 0 (v + v + at) / 2. So we get
1 2
0 0 0 0 0 2 x =x+(v +v +at)t / 2= x +v t+ at . This formula tells you how far a body
moves in time t if it moves with constant acceleration a, and if started at position x0 at
t=0 with speed v0 .
9. We can eliminate the time using (7), and arrive at another useful formula that tells us
what the final speed will be after the body has traveled a distance equal to
2 2
0 0 0 x−x after time t, v =v + 2a(x − x ).
10. Vectors: a quantity that has a size as well as direction is called a vector. So, for
example, the wind blows with some speed and in some direction. So the wind velocity
is a vector.
11. If we choose axes, then a vector is fixed by its components along those axes. In one
dimension, a vector has only one component (call it the x-component). In two
dimensions, a vector has both x and y components. In three dimensions, the
components are along the x,y,z axes.
12. If we denote a vector ( , ) then, cos , and sin . x y r􀁇= xy r =x=r θ r =y=r θ
Note that x2+y2=r2. Also, that tanθ =y/x.
14. Two vectors can also be added algebraically. In this case, we simply add the
components of the two vectors along each axis separately. So, for example,
Two vectors can be put together as (1.5,2.4)+(1,−1) = (2.5,1.4).