Honors Physics
9/17/13
Problem Solving
Note that the reason for solving problems in a physics course is to better understand important
concepts – that are often expressed mathematically. The math and numbers are secondary to
grasping basic ideas and how they are interrelated. To explain how a problem is being solved –
and demonstrate an appreciation of the underlying principles, use the following process.
1. Restate the problem in your own language by providing descriptive variable names to the
information given. Note the units provided and perform any unit conversions.
2. Identify what is to be found and give these unknowns descriptive names.
3. Draw a diagram showing how what is
given is related to what is to be found.
Use verbal explanations to clarify reasoning. Explain
why a particular relationship is being used, why a
particular intermediate result (distance, time, etc) is
being found, and so forth.
4. Translate the problem into the
language of mathematics by first
citing a basic idea/relationship and
describe how this specific problem illustrates this idea or relationship. Show the
connection between elements in this relationship and the specific variables in this
particular problem. Do not substitute numerical values yet
5. Set up an equation or a set of equations to be solved to find the unknowns listed above.
With the physical situation now modeled using mathematics – solve the equation(s) and
find values for the unknowns, using appropriate units and a reasonable number of
significant digits. It is best to solve problems using algebraic variables and obtain an
algebraic expression. Substitute numbers into equations only at the end. Note the
resulting numerical answers should always be expressed in meaningful units.
Example
To stop a car, you require first a certain reaction time to begin braking; then the car slows under
the constant braking deceleration. Suppose that the total distance moved by your car during
these two phases is 186 ft when its initial speed is 50 mi/hr, and 80 ft when the initial speed is
30 mi/hr. What are (a) your reaction time and (b) the magnitude of the deceleration?
Given: D1 = 186 ft = total braking distance from initial velocity v1 = 50 mph (73.3 ft/s)
D2 = 80 ft = total braking distance from initial velocity v2 = 30 mph (44.0 ft/s)
braking distance includes reaction time (v=constant) and deceleration (a=constant)
Find: (a) t = reaction time
(b) a = magnitude of acceleration while braking
Model:
assume both are the same
for the two trials
D
d
r
driver begins
to react
to = 0
braking starts
t = reaction
car stops
t+T
T = braking time
Honors Physics
9/17/13
r = motion at constant velocity = vo t (vo = initial velocity)
d = motion at constant acceleration: -a (a = magnitude only, “-” = direction)
Points in motion
begin to react
braking starts
time
0
t = (a)
position
velocity
acceleration
0
←
0
r = vo t
→
←
vo
Basic Idea during reaction:
kinematics of constant velocity to
relate r (part of D) to t and vo
Basic Idea during braking:
kinematics of constant acceleration to
relate d (part of D) to a and change in v
v 2 − vo2 = 2a(x − xo )
T = irrelevant, d = x - xo
x = xo + vt
t = needed
0 – vo2 = -2a (d)
-a = (b)
car stops
t+T
T=??
D=r+d
0
→
Δx = distance under constant acceleration = d
v = final velocity = 0
“-“ for negative acceleration so that value a is magnitude
r = 0 + vot
Combine two equations with D = r + d
D = r + d = vot + vo2/(2a)
Replace D and vo with two sets of braking distances and initial velocities and obtain two
equations relating given data to two unknowns: a and t.
(1) D1 = v1t + v12/(2a)
and
(2) D2 = v2t + v22/(2a)
Mathematics
Solve eqn (2) for 1/(2a) and substitute into eqn (1). Then solve for t in terms of given quantities.
1 D2 − v2 t
=
2a
v22
Substituting into (1) and solving for t:
The computing a:
a=
t=
v1
v2 2
v2
( )D
−( ) v
D1 −
1 v22
= 20.4 ft / s 2
2 D2 − v 2 t
v1 2
v2
2
= 0.74 s
2
Solution:
(a)
t = 0.74 s (a reaction time of a fraction of a second is reasonable for a driver to hit the
brakes)
(b)
a = 20 ft/s2 or 0.63 g (cars typically brake at rates less than g)