# Kinematicsx - Math-HS

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14 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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Kinematics deals with situations
involving motion.

S=distance/displacement.

If s=0, object is at starting point

+
-

relates to position from a certain point (above/right
vs

below/left).

V=velocity=rate at which distance changes over time

+
-

relates to direction of motion (forwards/up
vs

down/back)

If v=0, object is stationary

A=acceleration=rate at which velocity changes over time

+
-

relates to speeding up/slowing down.

If a=0, object is moving at constant speed

t=time

Initial values: when t=0

Differentiate

S

displacement

V

velocity

A

Acceleration

Integrate (add one to power/divide by new power)

For example…

2
3
4
2
6
3
2
2
t
t
v
t
t
dt
ds
v

10
3
6
3
6
)
6
(
2

v
The van’s velocity at the end
of the 6
th

second is
10metres/second.

More examples

1.
A stone is thrown vertically upwards into the air. Its height
above ground (in metres) is given by: s(t)=30t
-
5t
2

a)
Calculate the initial velocity

b)
Calculate the maximum height reached by the object

c)

2.
The velocity (m/s)of an object t seconds after it started from the
origin is given by:

v(t)=3t
2
-
14t
-
5

The object is travelling in a straight line in an East
-
West
direction.

a)
Give formulae for acceleration & distance after t seconds.

b)
Calculate the initial velocity. In which direction is the object moving?

c)
When is the object at rest?

d)
Calculate the minimum velocity. Interpret this answer.

e)
How far did the object travel in the 8
th

second?

1. A stone is thrown vertically upwards into the air. Its height above ground (in metres) is given by:
s(t)=30t
-
5t
2

Calculate the initial velocity

Calculate the maximum height reached by the object

a)
v=s’=30
-
10t

when t=0, v=30m/s

b)
Max height occurs when v=0

v=30
-
10t=0

t=3sec

s(3)= 30t
-
5t
2

=45m : max height is 45m.

c)
Object returns to ground when s=0

30t
-
5t
2

=0

5t(6
-
t)=0

t=0 or 6s

The object returns to the ground after 6 seconds

(t=0 is the initial time on the ground)

The velocity (m/s)of an object t seconds after it started from the origin is given by:

v(t)=3t
2
-
14t
-
5

The object is travelling in a straight line in an East
-
West direction.

Give formulae for acceleration & distance after t seconds.

Calculate the initial velocity. In which direction is the object moving?

When is the object at rest?

Calculate the minimum velocity. Interpret this answer.

How far did the object travel in the 8
th

second?

a)
a=v’=6t
-
14

s= ∫ 3t
2
-
14t
-
5.dt=3t
3
/3
-
14t
2
/2
-
5t+c.

s=t
3
-
7t
2
-
5t (since s=0 when t=0, c=0)

b)
V(0)= 3*0
2
-
14*0
-
5 =
-
5m/s

The object is moving west
-
>east

c)
The object is at rest when v=0

3t
2
-
14t
-
5=0

(3t+1)(t
-
5)=0

t=
-
1/3 or 5 : object is at rest after 5sec (
-
1/3 is before motion started)

d)
Minimum velocity occurs when a=0

a= 6t
-
14=0

t=14/6 sec (2.33sec)

V= 3t
2
-
14t
-
5

v(14/6)=
-
21.33m/s The minimum velocity is
-
21.33m/s

e)
During the 8
th

second, the object travels the difference of distance of the object
from the origin at t=7 and t=8.

At t=7, s(7)=7
3
-
7*7
2
-
5*7=
-
35m

At t=8, s(8)= 8
3
-
7*8
2
-
5*8=24m

the object moves from
-
35m to 24m during the 8
th

second, so travels a total of
59m.

(alternatively, use definite integration between t=7&8)

More examples

(try on your own then check

1.
A stone is thrown vertically upwards into the air. Its height
above ground (in metres) is given by: s(t)=30t
-
5t
2

a)
Calculate the initial velocity

b)
Calculate the maximum height reached by the object

c)

2.
The velocity (m/s)of an object t seconds after it started from the
origin is given by:

v(t)=3t
2
-
14t
-
5

The object is travelling in a straight line in an East
-
West
direction.

a)
Give formulae for acceleration & distance after t seconds.

b)
Calculate the initial velocity. In which direction is the object moving?

c)
When is the object at rest?

d)
Calculate the minimum velocity. Interpret this answer.

e)
How far did the object travel in the 8
th

second?

More examples

(try on your own then check

1.
A stone is thrown vertically upwards into the air. Its height
above ground (in metres) is given by: s(t)=30t
-
5t
2

a)
Calculate the initial velocity

b)
Calculate the maximum height reached by the object

c)

2.
The velocity (m/s)of an object t seconds after it started from the
origin is given by:

v(t)=3t
2
-
14t
-
5

The object is travelling in a straight line in an East
-
West
direction.

a)
Give formulae for acceleration & distance after t seconds.

b)
Calculate the initial velocity. In which direction is the object moving?

c)
When is the object at rest?

d)
Calculate the minimum velocity. Interpret this answer.

e)
How far did the object travel in the 8
th

second?

More examples

(try on your own then check

1.
A stone is thrown vertically upwards into the air. Its height
above ground (in metres) is given by: s(t)=30t
-
5t
2

a)
Calculate the initial velocity

b)
Calculate the maximum height reached by the object

c)

2.
The velocity (m/s)of an object t seconds after it started from the
origin is given by:

v(t)=3t
2
-
14t
-
5

The object is travelling in a straight line in an East
-
West
direction.

a)
Give formulae for acceleration & distance after t seconds.

b)
Calculate the initial velocity. In which direction is the object moving?

c)
When is the object at rest?

d)
Calculate the minimum velocity. Interpret this answer.

e)
How far did the object travel in the 8
th

second?

More examples

(try on your own then check

1.
A stone is thrown vertically upwards into the air. Its height
above ground (in metres) is given by: s(t)=30t
-
5t
2

a)
Calculate the initial velocity

b)
Calculate the maximum height reached by the object

c)

2.
The velocity (m/s)of an object t seconds after it started from the
origin is given by:

v(t)=3t
2
-
14t
-
5

The object is travelling in a straight line in an East
-
West
direction.

a)
Give formulae for acceleration & distance after t seconds.

b)
Calculate the initial velocity. In which direction is the object moving?

c)
When is the object at rest?

d)
Calculate the minimum velocity. Interpret this answer.

e)
How far did the object travel in the 8
th

second?

More examples

(try on your own then check

1.
A stone is thrown vertically upwards into the air. Its height
above ground (in metres) is given by: s(t)=30t
-
5t
2

a)
Calculate the initial velocity

b)
Calculate the maximum height reached by the object

c)

2.
The velocity (m/s)of an object t seconds after it started from the
origin is given by:

v(t)=3t
2
-
14t
-
5

The object is travelling in a straight line in an East
-
West
direction.

a)
Give formulae for acceleration & distance after t seconds.

b)
Calculate the initial velocity. In which direction is the object moving?

c)
When is the object at rest?

d)
Calculate the minimum velocity. Interpret this answer.

e)
How far did the object travel in the 8
th

second?

More examples

(try on your own then check

1.
A stone is thrown vertically upwards into the air. Its height
above ground (in metres) is given by: s(t)=30t
-
5t
2

a)
Calculate the initial velocity

b)
Calculate the maximum height reached by the object

c)

2.
The velocity (m/s)of an object t seconds after it started from the
origin is given by:

v(t)=3t
2
-
14t
-
5

The object is travelling in a straight line in an East
-
West
direction.

a)
Give formulae for acceleration & distance after t seconds.

b)
Calculate the initial velocity. In which direction is the object moving?

c)
When is the object at rest?

d)
Calculate the minimum velocity. Interpret this answer.

e)
How far did the object travel in the 8
th

second?