S1.3 Application: projectile motion
[A] The problem
Consider motion of projectile near the earth’s surface.
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- Specify initial conditions:
- launch at t=0 from x=0, y=0
- launch velocity:

- components:
- Identify assumptions:
- in words: acceleration is down, magnitude g
- in equations: ax=0 ay=-g
[B] Results
Analysis
Reveal
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Analysis
- Formulate strategy: think separately of x and y motions.
The horizontal motion
Set ax=0 and x0=0 in Key Point 1.4(a), (b)
vx=vx0+axt implies vx=vx0
implies x=vx0tThe vertical motion
Set ay=-g and use the y-forms of Key Point 1.4(a) and (b):
vy=vy0+ayt implies vy=vy0-gt
implies

The time of flight, tf
Flight ends on return to y=0
Thus set
with solutions![\[ 0= y=\xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{v_{y0}t - \frac{1}{2}gt^2} \]](mastermathpng-10.png)
![\[ \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{t=0}, \hspace{0.5cm}\mbox{\rm and}\hspace{0.5cm} \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{t= 2v_{y0}/g} \]](mastermathpng-11.png)
First solution identifies the start (launch)
Second solution must thus identify the end (crunch)
Setting vy0=v0sinθ deduce
tf=[2v0sinθ]/gCheck It!- Are the units OK?
- Does it make sense?
The range, R
Range is horizontal displacement in time tf. Thus
R=vx0tfSet
vx0=v0cosθand![\[ t_f=\xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{\frac{2v_{0}\sin \theta}{g}} \]](mastermathpng-12.png)
Then
Check It!- Are the units OK?
- Does it make sense?
The trajectory equation y(x)
Eliminating t between
x=vx0tand
![\[ y=\xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{v_{y0}t - \frac{1}{2}gt^2} \]](mastermathpng-14.png)
gives
![\[ y=v_{y0}\left[\frac{x}{v_{x0}}\right] - \frac{1}{2}g \left[\frac{x}{v_{x0}}\right] ^2 \]](mastermathpng-15.png)
Set
vx0=v0cosθandvy0=v0sinθ![$$y= x \tan \theta -[gx^2]/[ 2v_0^2 \cos ^2 \theta]$$](mastermathpng-16.png)
Check It!
- Are the units OK?
- Does it make sense?
Visualization:
Learning Resources
| HRW Chapter 4.5-6 | |

