Problem Solving Steps

1
Problem Solving Steps
1. Geometry drawing trajectory, vectors
, coordinate axes
free-body diagram, 2. Data
a table of known and unknown quantities,
including implied data. 3. Equations ( with
reasoning comments ! ), their solution in
algebraic form, and the final answers in
algebraic form !!! 4. Numerical calculations and
answers. 5. Check dimensional, functional,
scale, sign, analysis of the answers and
solution.
2
Formula Sheet PHYS
218 Mathematics p 3.14 1
rad 57.30o 360o/2p volume of sphere of
radius R V (4p/3)R3 Quadratic equation ax2
bx c 0 ?
Vectors and trigonometry


Calculus
3
Chapters 1 - 3 Constants g 9.80 m/s2,
Mearth 61024 kg, c 300 000 km/s, 1 mi
1.6 km 1-Dimensional Kinematics
3- or 2-Dimensional Kinematics
Equations of 1-D and 3-D kinematics for constant
acceleration
Circular motion
4
Exam Example 1 Coin Toss
Vy0
y
(problem 2.85)
Questions (a) How high does the coin go?
0
(b) What is the total time the coin is in the air?
Total time T 2 t 1.2 s
(c) What is its velocity when it comes back at
y0 ?
for y0 and vylt0 yields vy2 v02 ? vy -v0 -
6m/s
5
Exam Example 2 Accelerated Car (problems 2.7 and
2.17)
Data x(t)atßt2?t3, a6m/s, ß1m/s2, ? -2
m/s3, t1s Find (a) average and instantaneous
velocities (b) average and instantaneous
accelerations (c) a moment of time ts when the
car stops. Solution (a) v(t)dx/dt a2ßt 3?t2
v0a (b) a(t)dv/dt 2ß 6?t a02ß (c)
v(ts)0 ? a2ßts 3?ts20
0
x
V(t)
a
2ß
0
t
ts
a(t)
6
Exam Example 3 Truck vs. Car (problem 2.34)
Data Truck v20 m/s Car v00, ac3.2 m/s2
0
Questions (a) x where car overtakes the
truck (b) velocity of the car Vc at that x (c)
x(t) graphs for both vehicles (d) v(t) graphs
for both vehicles.
x
Solution trucks position xvt, cars position
xcact2/2

• xxc when vtact2/2 ? t2v/ac ? x2v2/ac
• (b) vcv0act ? vc2v

x
V(t)
truck
vc2v
car
car
truck
v
0
t2v/ac
t
t
0
v/ac
t2v/ac
7
Exam Example 4 Free fall past window (problem
2.84)
Data ?t0.42 s ? hy1-y21.9 m, v0y0, ay - g
y
Find (a) y1 (b) v1y (c) v2y
0
V0y0
1st solution
(b) Eq.(3) y2y1v1y?t g?t2/2 ? v1y -h/?t
g?t/2 (a) Eq.(4) ? v1y2 -2gy1 ? y1 -
v1y2 /2g -h2/2g(?t)2 h/2 g(?t)2 /8 (c)
Eq.(4) v2y2 v1y2 2gh (h/?t g?t/2)2
y1
ay
V1y
h
y2
V2y
2nd solution

• Free fall time from Eq.(3) t1(2y1/g)1/2 ,
  t2(2y2/g)1/2 ? ?tt1t2

(b) Eq.(4) ?
(c) Eq.(4) ?
8
Exam Example 5 Relative motion of free falling
balls (problem 2.94)
y
2
H
Data v01 m/s, H 10 m, ay - g
Find (a) Time of collision t (b) Position of
collision y (c) What should be H in order
v1(t)0.
0
1
Solution
(a) Relative velocity of the balls is v0
for they have the same acceleration ay g ? t
H/v0 (b) Eq.(3) for 2nd ball yields y H
(1/2)gt2 H gH2/(2v02) (c) Eq.(1) for 1st
ball yields v1 v0 gt v0 gH/v0 ,
hence, for v10 we find H v02/g
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Projectile Motion
ax0 ? vxv0xconst ay -g ? voy voy- gt x x0
vox t y yo voy t gt2/2 v0x v0 cos a0
v0y v0 sin a0 tan a vy / vx
Exam Example 6 Baseball Projectile
Data v022m/s, a040o
(examples 3.7-3.8, problem 3.12)
Find (a) Maximum height h (b) Time of flight
T (c) Horizontal range R (d) Velocity when
ball hits the ground
Solution
v0x22m/scos40o17m/s v0y22m/ssin40o14m/s

• vy0 ? h (vy2-v0y2) / (2ay) - (14m/s)2 / (-
  2 9.8m/s2) 10 m
• y (v0yvy)t / 2 ? t 2y / v0y 2 10m /
  14m/s 1.45 s T 2t 2.9 s
• R x v0x T 17 m/s 2.9 s 49 m
• vx v0x , vy - v0y

10
Exam Example 7 Ferris Wheel (problem 3.39)
Data R14 m, v0 3 m/s, a 0.5 m/s2

• Find
• Centripetal acceleration
• Total acceleration vector
• Time of one revolution T

Solution
(a) Magnitude ac a- v2 / r Direction to
center
?
(b)
(c)