- To solve projectile motion problems, perform the following steps:
- Determine a coordinate system. Then, resolve the position and/or velocity of the object in the horizontal and vertical components. The
components of positionsare given by the quantitiesxandy, and the components of the velocityvare given byvx=vcosθand
vy=vsinθ, wherevis the magnitude of the velocity andθis its direction.
- Analyze the motion of the projectile in the horizontal direction using the following equations:
Horizontal motion(ax= 0)
x=x 0 +vxt
vx=v 0 x=vx= velocity is a constant.
- Analyze the motion of the projectile in the vertical direction using the following equations:
Vertical motion(Assuming positive direction is up;ay= −g= −9.80 m/s^2 )
y=y 0 +^1
2
(v 0 y+vy)t
vy=v 0 y−gt
y=y 0 +v 0 yt−^1
2
gt^2
vy^2 =v 02 y− 2g(y−y 0 ).
- Recombine the horizontal and vertical components of location and/or velocity using the following equations:
s= x^2 +y^2
θ= tan−1(y/x)
v= vx^2 +vy^2
θv= tan−1(vy/vx).
• The maximum heighthof a projectile launched with initial vertical velocityv 0 yis given by
h=
v 02 y
2 g
.
• The maximum horizontal distance traveled by a projectile is called therange. The rangeRof a projectile on level ground launched at an angle
θ 0 above the horizontal with initial speedv 0 is given by
R=
v 02 sin 2θ 0
g.
3.5 Addition of Velocities
- Velocities in two dimensions are added using the same analytical vector techniques, which are rewritten as
vx=vcosθ
vy=vsinθ
v= vx^2 +vy^2
θ= tan−1(vy/vx).
- Relative velocity is the velocity of an object as observed from a particular reference frame, and it varies dramatically with reference frame.
- Relativityis the study of how different observers measure the same phenomenon, particularly when the observers move relative to one
another.Classical relativityis limited to situations where speed is less than about 1% of the speed of light (3000 km/s).
Conceptual Questions
3.2 Vector Addition and Subtraction: Graphical Methods
1.Which of the following is a vector: a person’s height, the altitude on Mt. Everest, the age of the Earth, the boiling point of water, the cost of this
book, the Earth’s population, the acceleration of gravity?
2.Give a specific example of a vector, stating its magnitude, units, and direction.
3.What do vectors and scalars have in common? How do they differ?
4.Two campers in a national park hike from their cabin to the same spot on a lake, each taking a different path, as illustrated below. The total
distance traveled along Path 1 is 7.5 km, and that along Path 2 is 8.2 km. What is the final displacement of each camper?116 CHAPTER 3 | TWO-DIMENSIONAL KINEMATICS
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