Starting with the kinematics equation x = vot + ½ at^2, derive an expresion relating the acceleration and initial velocity of an object to two paired position and time measurements. In other words, if an object's position is x1 at t1, find the acceleration a and the inital velocity vo in terms of x1, t2, x2, and t2. The derivation involves solving a system of two equations and two unknowns.
To derive an expression relating the acceleration and initial velocity of an object to two paired position and time measurements, we will start by rearranging the kinematics equation x = vot + ½ at² to solve for the initial velocity vo:
x = vot + (1/2)at²
Divide both sides of the equation by t:
x/t = vo + (1/2)at
Now, consider two position and time measurements, denoted as x1 at t1 and x2 at t2. Substituting these values into the equation, we get:
x1/t1 = vo + (1/2)at1
and
x2/t2 = vo + (1/2)at2
We have a system of two equations:
x1/t1 = vo + (1/2)at1
x2/t2 = vo + (1/2)at2
To solve this system of equations for vo and a, we can subtract the first equation from the second equation:
(x2/t2) - (x1/t1) = (vo + (1/2)at2) - (vo + (1/2)at1)
Simplifying:
(x2/t2) - (x1/t1) = 1/2 (at2 - at1)
Now, factoring out a common term of a:
(x2 - x1)/(t2 - t1) = 1/2 a(t2 - t1)
Multiply both sides by 2:
2(x2 - x1)/(t2 - t1) = a(t2 - t1)
Finally, solving for a:
a = 2(x2 - x1)/(t2 - t1)
Therefore, the expression relating the acceleration a and initial velocity vo to two paired position and time measurements x1 at t1 and x2 at t2 is:
a = 2(x2 - x1)/(t2 - t1)
To derive an expression relating the acceleration and initial velocity to two paired position and time measurements, we will follow these steps:
Step 1: Start with the kinematics equation x = vot + ½ at².
Step 2: Now, we'll use two paired position and time measurements: (x₁, t₁) and (x₂, t₂).
Step 3: For the first measurement (x₁, t₁), the equation becomes x₁ = vo*t₁ + (1/2)*a*t₁².
Step 4: For the second measurement (x₂, t₂), the equation becomes x₂ = vo*t₂ + (1/2)*a*t₂².
Step 5: We now have a system of two equations with two unknowns (vo and a).
Step 6: To solve this system, we can use a variety of methods like substitution, elimination, or matrices. I will explain the substitution method.
Substitution Method:
Step 7: Solve one equation for one variable.
From the first equation (x₁ = vo*t₁ + (1/2)*a*t₁²), we can isolate vo:
vo = (x₁ - (1/2)*a*t₁²) / t₁.
Step 8: Substitute the expression for vo into the second equation.
x₂ = [(x₁ - (1/2)*a*t₁²) / t₁] * t₂ + (1/2)*a*t₂².
Step 9: Simplify and rearrange the equation to solve for a.
Multiply both sides by t₁:
t₁*x₂ = (x₁ - (1/2)*a*t₁²) * t₂ + (1/2)*a*t₁*t₂².
Expand and rearrange the equation:
t₁*x₂ = x₁*t₂ - (1/2)*a*t₁²*t₂ + (1/2)*a*t₁*t₂².
Rearrange further:
t₁*x₂ - x₁*t₂ = (1/2)*a*t₁*t₂² - (1/2)*a*t₁²*t₂.
Factor out 'a':
t₁*x₂ - x₁*t₂ = (1/2)*a*t₁*t₂² - (1/2)*a*t₁²*t₂.
Now, simplify and factor out 'a':
(t₁*x₂ - x₁*t₂) = a * (1/2)*t₁*t₂² - (1/2)*t₁²*t₂.
Divide both sides by (1/2)*t₁*t₂² - (1/2)*t₁²*t₂:
a = (t₁*x₂ - x₁*t₂) / [(1/2)*t₁*t₂² - (1/2)*t₁²*t₂].
Step 10: Finally, we have the expression for acceleration (a) in terms of the position (x₁, x₂) and time (t₁, t₂) measurements.
To summarize, the expression for acceleration (a) in terms of position (x₁, x₂) and time (t₁, t₂) measurements is:
a = (t₁*x₂ - x₁*t₂) / [(1/2)*t₁*t₂² - (1/2)*t₁²*t₂].
You can use this expression to find the acceleration when given position and time measurements.