Physics 2514Lecture 17

P. Gutierrez

Department of Physics & AstronomyUniversity of Oklahoma

Physics 2514 – p. 1/14

Goals

Finish the discussion on projectile motion

Discuss the range formula;

Consider a couple of examples.

Physics 2514 – p. 2/14

Projectile Motion

Let’s consider how far a projectile travels;

Assume that it returns to the same height (y = y0)

Derive “Range Formula”

y − y0 = −g

2v2

0x

(x − x0)2 +

v0y

v0x

(x − x0)

0 = −g

2v2

0x

(x − x0)2 +

v0y

v0x

(x − x0)

(x − x0) =2

gv0xv0y =

v2

0sin(2θ)

g

2v0xv0y = 2v2

0sin θ cos θ = v2

0sin(2θ)

Physics 2514 – p. 3/14

Projectile Motion

Two comments on the range formula

Distance = (x − x0) =v2

0sin(2θ)

g

Maximum range when θ = 45◦

d

dθ

[

v2

0sin(2θ)

g

]

=

[

2v2

0cos(2θ)

g

]

= 0 ⇒ θ = 45◦

Sincesin(180◦ − φ) = sin φ ⇒ sin[2(90◦ − θ)] = sin(2θ)therefore the range is same for θ and 90◦ − θ

Physics 2514 – p. 4/14

Projectile Motion

Physics 2514 – p. 5/14

Projectile Motion

All objects fall at the same rate independent of there mass andhorizontal motion.

Physics 2514 – p. 6/14

Clicker

A 100 g golf ball rolls off a table and lands 2 m from the base ofthe table. A 200 g ball rolls off the same table with the samespeed. What distance from the base of the table does it land?(Ignore air resistance.)

1. < 1 m2. 1 m3. Between 1 and 2 m4. 2 m5. Between 2 and 4 m

Physics 2514 – p. 7/14

Example 1

A supply plane needs to drop a package of food to a personstranded on a desert island. The plane flies 100 m above theisland at a speed of 150 m/s. How far short of the target shouldit drop the package.

PSfrag replacements

150 m/s

100 m

D

Physics 2514 – p. 8/14

Solution

A supply plane needs to drop a package of food to a person stranded on a desert island.The plane flies 100 m above the island at a speed of 150 m/s. How far short of the targetshould it drop the package.

PSfrag replacements

v0xy0

g

xf

Knownx0 = 0 m y0 = 100 mv0x = 150 m/s v0y = 0 m/say = −g yf = 0 m

Unknownxf = ?

Time to hit ground: yf = 0 = −1

2gt2 + y0 ⇒ t =

√

2y0

g≈ 4.5 s

Horizontal distance traveled: xf = v0xt = v0x

√

2y0

g≈ 678 m

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Clicker

A supply plane needs to drop a package of food to a personstranded on a desert island. The plane flies 100 m above theisland at a speed of 150 m/s. What is the magnitude of thevelocity of the package just before it hits the ground?Recall: Time to hit ground: t =

√

2y0

g ≈ 4.5 s

A) 150 m/sB) 0 m/sC) 156 m/sD) 44 m/sE) 143 m/s

Physics 2514 – p. 10/14

Example 2

A rocket powered hockey puck has a thrust of 2 N and a totalmass of 1.0 kg. It is released rest on a frictionless table, 4.0 mfrom the edge of a 2 m drop. How far does the puck land fromthe base of the table?

PSfrag replacements

x

y

t0 t1

t2

Physics 2514 – p. 11/14

Example 2

A rocket powered hockey puck has a thrust of 2 N and a total mass of 1.0 kg. It isreleased rest on a frictionless table, 4.0 m from the edge of a 2 m drop. How far does thepuck land from the base of the table?

t0:x0 = −4 m y0 = 0 mv0x = 0 m/s v0y = 0 m/sax = F/m = 2m/s2 a0y = 0 m/s2

t1:x1 = 0 m y1 = 0 mv1x = ? m/s v1y = 0 m/sax = F/m = 2m/s2 a1y = 0 m/s2

t2:x2 = ? m y2 = −2 mv2x = ? m/s v2y = ? m/sax = F/m = 2m/s2 a2y = −9.8 m/s2

PSfrag replacements

x

y

t0 t1

t2

Physics 2514 – p. 12/14

SolutionPSfrag replacements

x

y

t0 t1

t2

x0 = −4 m x1 = 0 m

y1 = 0 m y2 = −2 m

v1y = 0 m/s ax = 2 m/s2

Velocity at t1: v2

1x − v2

0x = 2ax∆x ⇒ v2

1x = 2ax(x1 − x0) ⇒ v1x = 4 m/s

Time to hit ground (t = t2 − t1): y2 = −1

2gt2 + v1yt + y1 ⇒ y2 = −

1

2gt2

⇒ t =√

−2y2

g= 0.64 s

Horizontal distance traveled in time t: x2 = 1

2axt2 + v1xt + x1

⇒ (x2−x1) = 1

2axt2+v1xt = 2.96 m

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Assignment

Read sections 6.4

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