By algebraic manipulation of the first two kinematic equations for one-dimensional motion:
1) vf = vi + at
2) xf = xi + (vi)(t) + (1/2)(a)(t)^2
Obtain the other two kinematic equations :
3) vf^2 - vi^2 = 2ax
4) xf = xi + (1/2)(vi + vf)t
To obtain the other two kinematic equations, we need to manipulate the first two equations.
Starting with equation 1:
vf = vi + at
We square both sides:
(vf)^2 = (vi + at)^2
Expanding the right side using the binomial theorem:
(vf)^2 = (vi)^2 + 2vi(at) + (at)^2
Simplifying:
(vf)^2 = (vi)^2 + 2viat + a^2t^2
Next, let's subtract (vi)^2 from both sides:
(vf)^2 - (vi)^2 = 2viat + a^2t^2
Rearranging the terms:
(vf)^2 - (vi)^2 = 2at(vi + at)
We can replace (vi + at) with vf from equation 1:
(vf)^2 - (vi)^2 = 2at(vf)
This gives us equation 3:
vf^2 - vi^2 = 2at(vf)
Now, let's manipulate equation 2:
xf = xi + vit + (1/2)at^2
We can rearrange this equation:
xf - xi = vit + (1/2)at^2
Next, let's substitute vf from equation 1:
xf - xi = (vi + at)t + (1/2)at^2
Expanding and simplifying:
xf - xi = vit + at^2 + (1/2)at^2
Combining like terms:
xf - xi = vit + (3/2)at^2
Finally, let's replace vit with -xf + xi from the rearranged equation:
xf - xi = -xf + xi + (3/2)at^2
Rearranging the terms:
xf = xi + (1/2)at^2 + xf
This gives us equation 4:
xf = xi + (1/2)(vi + vf)t
So, the other two kinematic equations are:
3) vf^2 - vi^2 = 2at(vf)
4) xf = xi + (1/2)(vi + vf)t
To derive the other two kinematic equations using algebraic manipulation of the first two equations, let's start with equation 1:
vf = vi + at
Square both sides of equation 1 to eliminate the square root:
(vf)^2 = (vi + at)^2
Expand the right side using the distributive property:
(vf)^2 = (vi)^2 + 2(vi)(at) + (at)^2
Now, let's subtract (vi)^2 from both sides to isolate the terms involving vf:
(vf)^2 - (vi)^2 = 2(vi)(at) + (at)^2
Notice that we have a common factor of (at) on the right side. Factoring it out:
(vf)^2 - (vi)^2 = (at)(2vi + at)
Now, let's rearrange the equation to obtain the final form of equation 3:
(vf)^2 - (vi)^2 = 2avi + (at)^2
Equation 3 is:
vf^2 - vi^2 = 2avi + (at)^2
Now, let's move on to deriving equation 4 by manipulating equation 2:
xf = xi + (vi)(t) + (1/2)(a)(t)^2
We need to get rid of (vi)t on the right side to isolate xf:
xf - xi - (vi)(t) = (1/2)(a)(t)^2
Now, let's multiply both sides by 2 to eliminate the fraction:
2(xf - xi - (vi)(t)) = a(t)^2
Expand the left side using the distributive property:
2xf - 2xi - 2(vi)(t) = a(t)^2
Let's factor out (vi + vf) from the left side:
2xf - 2xi - 2(vi)(t) = 2(vi + vf)(t)
Now, divide both sides by 2:
xf - xi - (vi)(t) = (vi + vf)(t)
Rearrange the equation to get the final form of equation 4:
xf = xi + (1/2)(vi + vf)t
Equation 4 is:
xf = xi + (1/2)(vi + vf)t
Therefore, by manipulating equations 1 and 2, we derived equations 3 and 4:
3) vf^2 - vi^2 = 2avi + (at)^2
4) xf = xi + (1/2)(vi + vf)t