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The Philadelphia High School for Girls

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 Summer Science Assignment Ė 2009

 

You will be taking physics next year.  Students often have problems solving the equations.  This summer assignment will help you develop your equation solving skills.  I have tried to design it so it will be self explanatory.  Inevitably there will be problems I have not anticipated.  As students email me I will add new hints or sample problems, update this assignment and post it at www.geocities.com/richguffanti

Objective:
Students will be prepared to show the steps needed to solve equations similar to those below.  Do not just memorize the answers!
Students will be tested on similar problems during their first week back. 

Directions: 
a] Show the steps to solve the following equations.
b] Check your results with the answers below.
c] If you cannot figure out how to solve the equation copy your solution into an email and send it to me at RichGuffanti@Yahoo.com.
d] Submit the solutions to your physics teacher on the first day of
   school.

General Strategy

Often students find solving problems with all letters and no numbers confusing.  Here is a strategy
a] Substitute numbers in the equation for all but the letter you are solving for. 
b] Then solve the problem. 
c] Use this solution as a model for solving the original problem that had
   no numbers.  See Sample Problem A.


 

Sample Problem A

     Given:  S = D    Solve for D      S = Speed
            T                     D = Distance
                                  T = Time

Step 1:  Substitute 3 for S and 5 for T
         3 = D  
             5

Step 2:  Multiply both sides by 5
         35 = D 5      Note: The 5ís on the right cancel.
               5

         35 = D

Step 3:  Substitute S for 3 and T for 5
         ST = D

Sample Problem B

     Given:  S = D    Solve for T      S =  Speed
            T                     D = Distance
                                  T = Time
Step 1:  Multiply both sides by T
         ST = D T       Note: The Tís on the right cancel.
              T

         ST = D

Step 2:  Divide both sides by S
         ST = D         Note: The Sís on the left cancel.
         S    S

         T = D         
             S

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1    Given:  a = F    Solve for F      a = Acceleration
            m                     F = Force
                                  m = Mass
See Sample Problem A for a model.

2    Given:  a = F    Solve for m      a = Acceleration
            m                     F = Force
                                  m = Mass
See Sample Problem B for a model.

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3    Given:  I = V    Solve for V      V = Voltage
            R                     R = Resistance
                                  I = Current
See Sample Problem A for a model.

4    Given:  I = V    Solve for R      V = Voltage
            R                     R = Resistance
                                  I = Current
See Sample Problem B for a model.

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Sample Problem C

     Given:  A = BC2         Solve for C
            2          

Step 1:  Solve for C2: To isolate the C2 Multiply both sides by 2/B
 2 A = BC2 2            Note:  the B and 2 on the right cancel.
 B      2  B           

2A = C2 
 B
                                                    __
Step 2:  Take the square root of both sides:  Note:  √C2 = C
 ____
√2A/B = C

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5    Given:  25 = 17t2       Hint:  Solve for t2 then take the square root.
              2         Do not multiply/divide until the end.
Solve for t.            See Sample Problem C for a model.

6    Given:  X = at2         Hint:  Solve for t2 then take the square root.
            2                  X = Distance, a = Acceleration
Solve for t.            See Sample Problem C for a model.

7    Given:  KE = mV2        See Sample Problem C for a model.
              2         KE = Kinetic Energy
                        m = Mass
Solve for V.            V = Velocity
                        See Sample Problem C for a model.

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Sample Problem D

     Given:    _   Solve for L
A = 2π √L/B             Hint:  Square both sides.  Note (2π)2 = 4π2 
                               Then solve for L
Step 1:  Square both sides.
              ___  ___
AA = 22ππ √L/B √L/B

A2 = 4π2 L 
        B

Step 2:  Cancel 4π2/B on the right by multiplying both sides by B/4π2 
BA2 = L
4π2

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8    Given:    _   T = Time for the pendulum to swing back & forth once.
T = 2π √L/g    L = Length of the pendulum
              g = The Acceleration of Gravity

Solve for L   See Sample Problem D for a model.

9    Given:    _   g = The Acceleration of Gravity
3 = 2π √5/g   
             
Solve for g   Hint:  Square both sides.  Note (2π)2 = 4π2 

10   Given:    _   T = Time for the pendulum to swing back & forth once.
T = 2π √L/g    L = Length of the pendulum
              g = The Acceleration of Gravity

Solve for g   See Sample Problem D for a model.

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11   Given:  GMm   mV2            M = Planet Mass
        --- = --             m = Satellite Mass
         R2    R             R = Satellite's orbit radius
                             V = Satellite's velocity
Substitute 2πR for V         G = Universal Gravitational Constant
            T                T = Satelliteís time for 1 orbit.
                             Hint:  Divide both sides by m.
Solve for M                         Then solve for M.
                            
       Then Substitute 2πR for V 
                            
                        T

 


 

Problems with Subscripts

Because all the symbols have physical meaning, it is necessary to use subscripts.  Below there are 15 symbols that need subscripts.
Ex:  Vi = Initial Velocity           
     Vf = Final Velocity             
 
Some students find the subscripts difficult or confusing.  To help you with this the problems have been paired.  The first version looks like a standard Algebra 1 equation and the second version is the same equation using physics symbols.  The steps needed to solve both versions are the same.  Problems 22 to 25 you have to solve without this pairing.

 

Sample Problem E

     Given:      Vf - Vi                    a = Acceleration
        a = -------    Solve for Vf    t = Elapsed Time
               t                      Vi = Initial Velocity
                                      Vf = Final Velocity

Step 1:  Substitute A for a, B for Vf, C for Vi, D for t

Given:                                Solve for B
    B - C              
A = -------                      
      D

Step 1:  Multiply both sides by D     AD = B - C

Step 2:  Add C to both sides          C + AD = B

Step 3:  Substitute                   Vi + at = Vf
         Vi for C,
         a for A,
         t for D,
         Vf for B,

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Note:  If you have trouble with 12 - 19 see last page for common mistakes.

12   Given:
    B - C              
A = -------   Solve for C
      D

13   Given:                            a = Acceleration
    Vf - Vi                        t = Elapsed Time
a = -------    Solve for Vi       Vi = Initial Velocity
       t                          Vf = Final Velocity
Hint:  Use 13 as a model.

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14   Given:
    B - C              
A = -------    Solve for D
      D

15   Given:    Vf - Vi                  a = Acceleration
      a = -------    Solve for t  t = Elapsed Time
             t                    Vi = Initial Velocity
                                  Vf = Final Velocity
Hint:  Use 15 as a model.

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16   Given:  A = (B + C)D    Solve for C
               2

17   Given:  X = (Vi + Vf)t   X = Distance
                2       t = Elapsed Time
                        Vi = Initial Velocity
Solve for Vf            Vf = Final Velocity

Hint:  Use 18 as a model.

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18   Given:  A = (B + C)D      Substitute B + ED for C and solve for A
               2

19   Given:  X = (Vi + Vf)t    Substitute Vi + at for Vf and solve for X.
                2
Hint:  Use 19 as a model.

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Sample Problem F

     Given:                                    
AB + CD = AE + CF                          Solve for E
                                          
Step 1:  Subtract CF from both sides.      AB + CD = AE + CF
                                              - CF        CF
                                           AB + CD - CF = AE

Step 2:  Factor out C                      AB + C(D - F) = AE

Step 3:  Divide by A                       B + C(D - F) = E
                                               A

20   Given:                       M1 & M2 = Masses of Object 1 & 2
M1V1 + M2V2 = M1V1' + M2V2'    V1 & V2 = Velocities Before Collision
                             V1' & V2' = Velocities After Collision
Solve for V1'                Hint:  Use Sample Problem F for a model.
                                  M1 = A, & M2 = B
                                  V1 = C, & V2 = D
                                  V1'= E, & V2'= F


 

Sample Problem G

Light

     Given: 1/A = 1/B + 1/C         f = Focal Length of a Lens
                               Do = Distance to Object
Solve for A                    Di = Distance to Image
                               Hint:  Multiply both sides by A B C
Step 1:  Multiply both sides by ABC
ABC = ABC + ABC        
 A     B     C

BC = AC + AB

Step 2:  A is a common factor on the right side.
BC = A(C + B)

Step 3:  Divide both sided by (C + B)
  BC   = A
C + B

21   Given: 1/f = 1/Do + 1/Di        f = Focal Length of a Lens
                               Do = Distance to Object
Solve for f                    Di = Distance to Image
                               Hint:  Multiply both sides by f Do Di

22   Given: 1/f = 1/Do + 1/Di        f = Focal Length of a Lens
                               Do = Distance to Object
Solve for Do                    Di = Distance to Image
                               Hint:  Multiply both sides by f Do Di

23   Given: 1/f = 1/Do + 1/Di        f = Focal Length of a Lens
                               Do = Distance to Object
Solve for Di                    Di = Distance to Image
                               Hint:  Multiply both sides by f Do Di 


 

Answers

1    F = ma                            2    m = F/a

3    V = IR                           

4    R = V/I

5         _______                     
t = √2●25/17 = 1.71              

6         _____
t = √2X/a

7         ______
V = √2KE/m

8    L = gT2                
    4π2                 

9    g = 4π25                         
     32                          

10   g = 4π2L
     T2

11   M = RV2 = 4π2R3 
     G     GT2 

12   C = B - AD                       

13   Vi = Vf - at

14       B - C                        
D = -------                      
      A

15   t = Vf - Vi 
       a

16   C = 2A - B                       
     D                           

17   Vf = 2X - Vi
      t

18   A = BD + ED2                     
          2                      

19   X = Vit + at2 
          2  

20   V1í = V1 + M2 (V2 - V2í) 
          M1                 

21   f =   DiDo              
    Di + Do

22   Do =   fDi              
     Di - f    

23   Di =   fDo              
     Do - f