Collision Calculator is a software that solves one dimensional collision problem of two objects. It performs the following calculations. When two initial velocities are known, get two final velocities after collision. When two final velocities are known, get two initial velocities before collision. When two initial velocities and one final velocity are known, get another final velocity and coefficient of restitution COR).

It's possible to change one of input data one after another. Thus it's possible to get output data which correspond to changed input data.

（Example)

A first ball of mass 1 Kg is moving with the velocity 1 m/s, and second ball of mass 2 Kg is stationary. Calculate the velocities of both ball after collision.

When the first ball velocity is changed to 0.9, 0.8, and 0.7 m/s, calculate the velocities of both ball after collision.

Assume that coefficient of restitution is 0.5.

(Answer)

Enter as follows,

e = 0.5

M1 = 1, V1 = 1

M2 = 2. V2 = 0

Touch out P button that corresponds to the data which is planning to be changed.

We are planning to change the velocity of the first ball.

Touch out V1_P button.

We obtain 0.0 m/s and 0.5 m/s as the final velocities.

Delta data is attached to the V1 data as an increment.

Enter as follows.

Delta = 0.1

Touch out - button of the V1 data. V1 data is decremented.

We obtain 0.0 m/s and 0.45 m/s as the final velocities that correspond to the first ball velocity 0.9 m/s.

Next touch out - button. We obtain the next output data.

2. Data

e: Coefficient of restitution

M1: Mass of first body.

M2: Mass of second body.

V1: Initial velocity of first body.

V2: Initial velocity of second body.

V1': Final velocity of first body.

V2': Final velocity of second body.

Delta (not labeled): Increment attached to P button data.

3. Equations

Following are equations for collision.

Calculation is to solve simultaneous equations.

Law of momentum conservation

M1*V1 + M2*V2 = M1*V1' + M2*V2'

Equation of restitution

e = -(V1' - V2')/(V1 - V2)

e: Coefficient of restitution

e = 0: Perfectly inelastic collision

0 < e < 1: Elastic collision

e = 1: Perfectly elastic collision

1 < e: Super elastic collision.

4. Buttons and Calculations

4.1. P buttons

P button assigns input data for calculation. When P button is touched out, software reads data from the P button data field and executes calculation. The calculation is usually to solve multiple equations by referencing other data. Number of input data equals one. Number of output data equals the number of equations used for the calculation.

When the data field is blank, it is interpreted as illegal data.

When the data field is illegal character, it is interpreted as zero value.

Following calculations are executed.

e_P button

M1_P button

M2_Pbutton

V1_P button

V2_P button

V1' = ((M1*V1 + M2*V2) + e*M2*(V2 - V1))/(M1 + M2)

V2' = ((M1*V1 + M2*V2) + e*M1*(V1 - V2))/(M1 + M2)

4.2. + and - buttons

Delta data is attached to the P_button data as an increment.

Before calculation, P button data is incremented or decremented by the + or - button.

For example, when + or - button of V1 data is touched out, below is carried out first.

V1 = V1+ Delta

V1 = V1 - Delta

Next the P button calculation is performed.

By this buttons, it's possible to change one of input data one after another and to get output data which corresponds to changed input data.

4.3. G buttons

G button assigns output data for calculation. When G button is touched out, software reads data required by the calculation as input data, and executes calculation.

Following calculations are executed.

V1'_G button

V1' = V1 + (M2/M1)*(V2 - V2')

e = -(V1' - V2')/(V1 - V2)

V2'_G button

V2' = V2 + (M1/M2)*(V1 - V1')

e = -(V1' - V2')/(V1 - V2)

4.4. C button

V1'_C button

V1'_C button clears initial state data.

When it was pushed over 3 seconds, it clears e data.

V2'_C button

V2'_C button clears final state data.

When it was pushed over 3 seconds, it clears e data.

4.5. Color Change

P button and G button execute calculation and generate output data from input and referenced data.

Color change shows what data are input, output, or referenced data in this button calculation.

Input data: Blue color

Referenced data: Yellow color

Output data: Green color

Uncalculated data: Red color

Uncalculated data is an output data that is not normally calculated. This uncalculated data is not output. The previous data is remains unchanged in this data field.

5. Notes and Restrictions

5.1. Data unit

Data unit of mass has to be same in first and second mass data.

Data unit of velocity has to be same in all velocity data.

5.2. Coefficien of Restitution (COR)

Range of restitution value is usually from 0.0 to 1.0.

This software solves simultaneous equations without limiting the range. Therefore it is possible to get coefficient of restitution data that is outside of the above range by input data.

6. Examples

6.1. Example -1

A first ball of mass 1 Kg collides with the second ball of mass 2 Kg.

After collision the first ball is moving with the velocity 1 m/s, and second ball is moving with the velocity 2 m/s. Calculate the velocities of both ball before collision.

When the first ball velocity after collision is 0.9, 0.8, and 0.7 m/s, Calculate the velocities of both ball before collision.

Assume that coefficient of restitution is 0.5.

(Answer)

Enter as follows,

e = 0.5

M1 = 1, V1' = 1

M2 = 2. V2' = 2

Next we want to change the velocity V1' .

Touch out V1'_P button.

We obtain 3.0 m/s and 1.0 m/s as the final velocities.

Delta is attached increment data to the V1 data.

Enter as follows.

Delta = 0.1

Touch out - button of the V1' data.

We obtain 3.1 m/s and 0.9 m/s as the initial velocities when the first ball velocity after collision is 0.9 m/s.

6.2. Example -2

A first ball of mass 1 Kg is moving with the velocity 1 m/s, and second ball of mass 2 Kg is stationary. After collision the first ball velocity is - 0.2 m/s.

Calculate the second ball velocity and coefficient of restitution.

(Answer)

Enter as follows,

M1 = 1, V1 = 1, V1' = -0.2

M2 = 2. V2 = 0

Touch out V2'_G button.

We obtain 0.6 m/s and 0.8 as the second ball velocity after collision and coefficient of restitution, respectively.

6.3. Example -3 Super elastic collision

We assume two special balls. A first ball of mass 1 Kg is moving with the velocity 1 m/s, and second ball of mass 1 Kg is stationary. Calculate the velocities of both balls after collision when the coefficient of restitution(COR) is the following respective values.

1.5, 1.4, 1.3, .... , -1.3, -1.4, -1.5.

(Answer)

Enter as follows,

e = 1.5

M1 = 1, V1 = 1

M2 = 1. V2 = 0

Touch out e_P button

Delta data is attached to the e data as an increment.

Enter as follows.

Delta = 0.1

Touch out - button of the e data, and repeat it.

We obtain the following results.

e V1' V2'

1.5 -0.25 1.25

1.4 -0.20 1.20

1.3 -0.15 1.15

1.2 -0.10 1.10

1.1 -0.05 1.05

1.0 0.00 1.00　perfectly elastic collision

0.9 0.05 0.95

0.8 0.10 0.90

0.7 0.15 0.85

0.6 0.20 0.80

0.5 0.25 0.75

0.4 0.30 0.70

0.3 0.35 0.65

0.2 0.40 0.60

0.1 0.45 0.55

0.0 0.50 0.50 perfectly inelastic collision

-0.1 0.55 0.45

-0.2 0.60 0.40

-0.3 0.65 0.35

-0.4 0.70 0.30

-0.5 0.75 0.25

6.4. Example -4 Required bat speed change for COR change.

A ball of COR 0.44 and mass 145 g was used by a baseball game. A pitcher threw the ball with the speed 120 km/h. A batter swung a bat of mass 900 g by the speed 100 km/h. Calculate the ball speed after ball- bat collision.

The ball was changed to a new ball of COR 0.41 of the same mass. Calculate the new bat speed for the new ball to reach the previous ball arrival point.

(Answer)

Enter as follows,

e = 0.44

M1 = 145, V1 = 120

M2 = 900. V2 = -100

Touch out e_P button

We obtain the ball speed 152.84 km/h after ball- bat collision.

Delta data is attached to the e data as an increment.

Enter as follows.

Delta = 0.01

Touch out - button of the e data three times.

When the COR is changed to 0.41, the ball speed after collision is changed to 147.16 km/h.

We approximate V2 data that gives the previous ball speed 152.84 km/h by changing the V2 data. The V2 data can be changed by Delta data and + /- butttons.

Following shows that the V2 data (= bat speed) 104.68 km/h gives the previous ball speed 152.84 km/h.That is the bat speed 104.68 km/h hits the ball to the previous arrival point.