Starting from the constant-acceleration kinematic equations, write a formula that gives vf in terms of t, xi, xf, and a.
Starting from the constant-acceleration kinematic equations, write a formula that gives t in terms of xi, xf, vi, and vf.
To write the formula for vf (final velocity) in terms of t (time), xi (initial position), xf (final position), and a (acceleration), we can start with one of the constant-acceleration kinematic equations:
xf = xi + vit + (1/2)at^2
Rearranging this equation to solve for vf, we get:
xf - xi = vit + (1/2)at^2
Since vi (initial velocity) is not mentioned in the question, we can assume it is zero (vi = 0). This simplifies the equation to:
xf - xi = (1/2)at^2
Now, we can multiply both sides of the equation by 2 to eliminate the fraction:
2(xf - xi) = at^2
Finally, we can take the square root of both sides of the equation to solve for vf:
√(2(xf - xi)/a) = t
Therefore, the formula for vf in terms of t, xi, xf, and a is:
vf = √(2(xf - xi)/a)
Now let's move on to the formula for t (time) in terms of xi, xf, vi (initial velocity), and vf (final velocity). We'll start with another constant-acceleration kinematic equation:
xf = xi + vit + (1/2)at^2
Rearranging this equation to solve for t, we get:
xf - xi = vit + (1/2)at^2
Now, let's isolate the term involving t:
xf - xi - vit = (1/2)at^2
Multiplying both sides of the equation by 2 to eliminate the fraction:
2(xf - xi - vit) = at^2
Dividing both sides by a:
2(xf - xi - vit)/a = t^2
Taking the square root of both sides, we obtain:
√(2(xf - xi - vit)/a) = t
Therefore, the formula for t in terms of xi, xf, vi, and vf is:
t = √(2(xf - xi - vit)/a)
Note that in this formula, the initial velocity vi is included, unlike in the previous formula, since it is required to calculate time.