CK-12-Physics - Intermediate

http://www.ck12.org Chapter 3. Two-Dimensional Motion

V, except to say thatVmust be is at least some minimum value if the intended target is to be reached.

Since 45◦gives the greatest range, what kind of range will smaller and larger angles than 45◦give? Forty-five

degrees is an optimum condition and we notice that forθ= 30 ◦andθ= 60 ◦(equally distributed about 45◦we have

2 θ= 2 ( 30 ◦) = 60 ◦,and 2 θ= 2 ( 60 ◦) = 120 ◦which gives sin 60◦=sin 120◦. The implication is that the range is

the same for angles( 45 ◦+θ)and( 45 ◦−θ). This condition can be readily proven with a bit of trigonometry: Is

sin( 2 ( 45 ◦+θ)) =sin( 2 ( 45 ◦−θ))an identity? The distribution gives sin( 90 ◦+ 2 θ) =sin( 90 ◦− 2 θ), which after

expansion gives: cos( 2 θ) =cos( 2 θ)and confirms the statement is an identity.

Lastly, we consider expressingyas a function ofx, rather thant.

By solving equation 1 fortand substituting the result into equation 2, we have, after recalling tanθ=cossinθθ,

yf=

1

2

g

(∆x)^2

(vcosθ)^2

+∆xtanθ+yi

IfXi=0 then∆xabove can be replaced withX, orR.

1. A quantity that has both magnitude and direction (as velocity does) is a vector quantity.


2. Inertial reference frames are constant velocity frames and are equivalent to each other.


3. The relative velocity of one object compared to another can be computed using vector addition once the
   velocities of each object in their respective reference frame is known. Both frames have their velocities
   referenced to an “at-rest” reference frame.


4. Vectors can be added graphically from head to tail and numerically by adding all thex−components of each
   vector together and all they−components of each vector together.


5. Projectile motion can be analyzed by considering independently thex−andy−components of the motion of
   the projectile.