Marathon

Two racers are running marathon (42 m) by running 21km in one direction and running back to the place where they started . One racer run with average sped 18 km/h, and the other with speed 14 km/h. How far from the start is second racer when they meet? How much later will second racer come to the finish?

Solution:

v₁=18 km/h
v₂=14 km/h
A... place where slower runner will be when faster runner rich 21 km
B... place of meeting

C... place where direction of running changes

t₁=D/v₁=1,17 h ... time when faster runner rich 21 km
s_A=v₂t₁=16,33 km ... distance which slower runner has pass during time t₁
s_AC=s_AB+s_BC=D-s_A=4,67 km ... path that slower runner has to pass till point C
s_AC=v₁t_x+v₂t_x
where t_x is time which pass till the meeting point B

t_x=s_AC/(v₁+v₂)=0,146 h
s_AB=t_xv₂=2,04 km

Now, distance of the slower runner from the start in the moment of meeting is

s_A+s_AB=2,04 km + 16,33 km =18,37 km

Faster runner need 42km/v₁=2,332 h, while slower runner need 42/v₂=3h to complete the marathon.

So slower runner will rich finish 3h-2,333h=0,67h=40,2 min=2412 sec later

Speed of light

Calculate how much time does light need to arrive from sun to the earth. (c=3x10⁸ m/s, average distnace between sun and earth is d=1,49 x10¹¹ m)

Solution:

t=d/c=500 sec= 8 min 20

Free fall

During the last second of free fall object pass half of the full length of falling path. From which height and how long did the object fall?

Solution:
h,t...full falling path,time
h₂,t₂...path,time in last second
h₁...rest of path

h₁+h₂=h
h₁=h₂=h/2

Simple acceleration example

One car can accelerate from 0 to 100 kmh^-1 in 9 sec. Calculate acceleration and path if acceleration is constant.

v₁=0 kmh^-1
v₂=100 kmh^-1 =27,78 ms^-1
t=9,0 sec

Solution:



where s is distance of car from starting position.

In reality car does not accelerate constantly, and with this example we calculated mid acceleration.

Meeting problem

In 17h 33 min train In Paris (France) started moving toward Madrid (Spain) with velocity 140 km/h, while second train in 18h 0min started moving from Madrid toward Paris with velocity 120 km/h. Distance between Paris and Madirid is 1100 km. When and where will theese trains meet?

d=1100 km
v₁=140 km/h
v₂=120 km/h
t₁=17h 33min
t₂=18h 00min

Solution:

Trains will meet in time t far s₁ from Paris. In the meeting time first train will pass s₁=v₁(t-t₁), while the second train is far from initial position s₂=v₂(t-t₂).
These two paths together make distance d: s₁+s₂=d whic makes
v₁t-v₁t₁+v₂t-v₂t-v₂t₂=d












t=17h 33 min+ 4h 26 min = 21h 59 min

Place of meeting is far from Paris:
s₁=v₁(t-t₁) =621,32 km

Overtake problem

One truck which is long 6m, velocity 80 km/h started overtaking another truck which is 15m long and with velocity 65 km/h. How long does overtaking last, and how far is faster truck from initial postition if overtaking starts and ends when two trucks are at distance 30m?

Solution:
d₁=6m
v₁=80 km/h
d₂=15m
v₂=65 km/h
d=30 m












When slower truck pass path s₂=v₂t, faster car pass:
s₁=v₁t=d+d₂+v₂t+d+d₁
from which we can calculate t,





Faster car from his initial position pass:

s₁=v₁t=432 m

We can see that if relative speed v₁-v₂ is less overtaking time is growing.

Linear motion

When object is displacing during a time interval average velocity is described by formula





If we want to calculate instantaneous velocity at one point we must make time interval small (infinite small)





SI unit for velocity is meters/seconds (ms^-1).
In the case of linear motion and constant velocity we can write formula (1) as





where s is path traveled during time t. We also have






Acceleration is defined as rate of change of velocity








In SI units, acceleration is measured in meters/seconds² (ms-²)

Special way of motion is when a=const. in which case we have:

v=v₀+at,

s=v₀t+at²/2,

v²=2as+v₀²,

where v₀ is starting speed and path s is measured from time t=0. When object is speeding up value of a is positive and if object is slowing down value of acceleration 'a' is negative.
In free fall which is one of the important movements where a=const. acceleration is labeled as g and in numerical problems we calculate as g=9,81 ms^-2.
Displacement, velocity and acceleration are vectors which because of simplicity they were written as scalars.