Question

A. GALILEO’S EQUATIONS FOR MOTION

SHOW THE ALGEBRA STEPS TO REACH EQUATIONS 9, 10, 11, 12

1. Velocity: v = (x₂- x₁)/(t₂- t₁),  which may be written v = Δx/Δt  [equation 1]

2. Acceleration: a = (v₂- v₁)/(t₂- t₁), which may be written a = Δv/Δt [equation 2)

3. Average velocity: <v> =(v₁+v₂)/2  [equation3]

4.  <v> = d/t   [equation 4]

5. COMBINING equations 3 and 4: (v₁+v₂)/2 = d/t  [equation 5]

The equations below are results valid only for the case of constant acceleration.

6. Velocity: v = d/t    [equation 6]

7. Acceleration: a = (v - v_o)/t     [equation 7]

SHOW THE ALGEBRA STEPS FOR REACHING THE FOLLOWING EQUATIONS:

8. Use equation 5 written for the "initial" and "final" velocities vo and v. Then solve the equation for d:

9.    d = (v_o+ v)t/2     [equation 9]

10. Solve equation 7 for v:   v = v_o+ at      [equation 10]

11. Replace v in equation 9 with v_o+ at (which equals v): d = v_ot + (at²/2)  [equation 11]

12. Finally, multiply equation 7 by equation 9, and rearrange to get:

v²= v_o² + 2ad    [equation 12]

Answer 1

8] Average velocity is given by:

now, substitute v₁ = v_o and v₂ = v to get,

re-arranging this gives, (equation 9)

but from above (equation 7), the acceleration a is related to initial and final speed by:

=> at = v - v_o

=> (equation 10)

substitute this in d to get,

=> (equation 11)

now, consider the equation for acceleration,

this can be written as,

substitute this in to get,

=>

=>

=>

=>

re-arrange this to get,

(equation 12).