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Projectile Motion Examples

Example 4.3 The Long Jump

Problem A long-jumper (Fig. 4.12) leaves the

ground at an angle ?i 20 above the horizontal

at a speed of vi 8.0 m/s. a) How far does

he jump in the horizontal direction?

(Assume his motion is equivalent to that of a

particle.) b) What is the maximum height

reached?

Example Driving off a cliff!!

- y is positive upward, yi 0 at top. Also vyi

0 - How fast must the motorcycle leave the cliff to

land at - xf 90 m, yf -50 m? vxi ?

vx vxi ? vy -gt x vxit, y - (½)gt2 Time

to Bottom t v2yf/(-g) 3.19 s vxi (xf/t)

28.2 m/s

Kicked football

- ?i 37º, vi 20 m/s
- ? vxi vicos(?i) 16 m/s, vyi visin(?i) 12

m/s - a. Max height? b. Time when hits ground?
- c. Total distance traveled in the x direction?
- d. Velocity at top? e. Acceleration at top?

vf

vyi

vxi

Conceptual Example

vyi

- Demonstration!!

vxi

vyi

vi

vxi ?

Conceptual Example Wrong Strategy

- Shooting the Monkey!!
- Demonstration!!

vi ?

Example

- Range (R) of projectile ? Maximum horizontal

distance before returning to ground. Derive a

formula for R.

xi 0 yi 0

?i

?i

?i1

?i1

?i2

- Range R ? the x where y 0!
- Use vxf vxi , xf vxi t , vyf

vyi - gt - yf vyi t (½)g t2, (vyf) 2

(vyi)2 - 2gyf - First, find the time t when y 0
- 0 vyi t - (½)g t2
- ? t 0 (of course!) and t (2vyi)/g
- Put this t in the x formula xf vxi (2vyi)/g

? R - R 2(vxivyi)/g, vxi vicos(?i), vyi visin(?i)

- R (vi)2 2 sin(?i)cos(?i)/g
- R (vi)2 sin(2?i)/g (by a trig identity)

Example 4.5 Thats Quite an Arm!

Problem A stone is thrown from the top of a

building at an angle ?i 26 to the horizontal

and with an initial speed vi 17.9 m/s, as in

Fig. 4.14. The height of the building is 45.0

m. a) How long is the stone "in

flight"? b) What is the speed of the

stone just before it strikes the

ground?

Example A punt!

- vi 20 m/s, ?i 37º
- vxi vicos(?i) 16 m/s, vyi visin(?i) 12 m/s

Proof that projectile path is a parabola

- xf vxi t , yf vyi t (½)g t2
- Note The same time t enters both equations!
- ? Eliminate t to get y as a function of x.
- Solve x equation for t t xf/vxi
- Get yf vyi (xf/vxi) (½)g (xf/vxi)2
- Or yf (vyi /vxi)xf - (½)g/(vxi)2(xf)2
- Of the form yf Axf B(xf)2
- A parabola in the x-y plane!!

Problem

vi 65 m/s

65

Example 4.6 The Stranded Explorers

Problem An Alaskan rescue plane drops a package

of emergency rations to a stranded party of

explorers, as shown in the picture. If the plane

is traveling horizontally at vi 42.0 m/s at a

height h 106 m above the ground, where does the

package strike the ground relative to the point

at which it is released?

vi 65 m/s

h