Particle Velocity

If a particle is moving it will have a certain velocity.  The velocity of the particle could be constant with in a certain period of time, or it could change within a certain period of time.  For right now I am only going to talk about a constant velocity.

As with position of a particle, the velocity of a particle is defined by an observer, and what the velocity actually is can be subjective to what the observer is actually seeing.  For example, as a car is moving down the freeway an observer on earth would say that car is going at a certain constant speed, such as 60 mph.  However, what if I told you that car is moving faster than 60 mph.  It is actually moving at the speed the earth is spinning, plus the speed that the earth is revolving around the sun, plus the speed that the solar system is moving through the galaxy, and so on.  But since the observer on earth can’t see this they say the car is moving at 60 mph.  Remember, humans cannot feel an object that is moving at a constant velocity, we can only visually observe constant velocity.  If you say that you can feel velocity you are actually feeling a change in velocity.

Depending on the what the particles velocity is the particle position will change  within a given time.  This can be expressed by the equation below.

(Eq 1)  $v=\frac{ds}{dt}$

ds = Change in displacement

dt = Change in time

Example

For example a particle travels a distance of 100m in 20s what was its velocity?The particle is not accelerating.

Solution

$v = \frac{100m}{20} = 5 \frac{m}{s}$

Next due to the fact that we live in a 3-dimensional world the velocity of a particle could be expressed using the Cartesian Coordinate system where there will be certain velocity in the x-direction, in the y-direction, and in the z-direction. Expressing velocity like this doesn’t give us an easy value to work with.  However, we can find the magnitude of the these 3 values using the equation below.

(Eq 2)  $v=\sqrt{v_xi^2+v_yj^2+v_zk^2}$

Example

The velocity of a particle has the following x, y, and z values, 2i$\frac{m}{s}$, 4j$\frac{m}{s}$, 7k$\frac{m}{s}.  What is the total velocity of the particle?

Solution

$\sqrt{(2i)^2+(4j)^2+(7k)^2}=8.3\frac{m}{s}$

Above I have only talked about velocity.  However, as mentioned above, velocity can be used to define the position of an object in respect to time.  In order to do this you would have to use the following equation.

(Eq 3) $s=vt+s_o$

$s$ = final position

$v$ = velocity

$t$ = time

$s_o$ = initial position

Example

A train is 45 miles away from the station.  It is moving at a constant speed of 50mph.  The current time is 3 pm.  What time will the train arrive at the station.  Assume the train does not accelerate or decelerate.

Solution

$ v = 50mph$, $s_0 = o$,  $s = 45 miles$

Modify equation 3 to the following.

$t=\frac{s-s_o}{v}=\frac{45~miles}{50~mph}=0.9~hours$

0.9 hours can be converted to 54 minutes, which means the train will arrive at 3:54 pm

 

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