How do you derive the equation for position vs. time from the velocity vs. time equation? For example let's say the velocity equation is v = 0.17t + -0.43
How would I go about figuring out the position vs. time equation with accurate numerical constants?
To derive the equation for position vs. time from the given velocity equation, you need to integrate the velocity equation with respect to time. Here is a step-by-step process to find the position vs. time equation from the given velocity equation v = 0.17t - 0.43:
Step 1: Start with the given velocity equation:
v = 0.17t - 0.43
Step 2: Integrate both sides of the equation with respect to time to find the position equation:
∫v dt = ∫(0.17t - 0.43) dt
Step 3: Integrate each term on the right side of the equation separately:
∫v dt = ∫(0.17t) dt - ∫(0.43) dt
Step 4: Integrate each term using the power rule of integration:
∫v dt = 0.17∫t dt - 0.43∫dt
Step 5: Simplify the integrals:
∫v dt = 0.17 * (t^2/2) - 0.43 * t + C
Step 6: Rewrite the integral on the left side as the position equation:
∫v dt = ∫(dv/dt) dt = ∫dx = x
Step 7: Replace the integral of velocity with the position variable:
x = 0.17 * (t^2/2) - 0.43 * t + C
Therefore, the position vs. time equation for the given velocity equation v = 0.17t - 0.43 is:
x = 0.085t^2 - 0.43t + C
where C is the constant of integration, representing the initial position at t = 0.
To derive the position vs. time equation from the velocity vs. time equation, you need to integrate the velocity equation with respect to time. By integrating, you can find an expression that represents the relationship between position and time.
In this example, the velocity equation is given as v = 0.17t - 0.43.
Step 1: Integrate the velocity equation
To integrate v with respect to t, you treat v as the derivative of the position equation with respect to time. Integrating gives you:
∫v dt = ∫(0.17t - 0.43) dt
Step 2: Evaluate the integral
Integrating 0.17t with respect to t gives (0.17/2)t^2 = 0.085t^2. Similarly, integrating -0.43 with respect to t gives -0.43t.
∫v dt = 0.085t^2 - 0.43t + C
Step 3: Find the value of the constant
In the integration process, a constant of integration (C) appears. To find its value, you need additional information such as the initial position or a specific position at a given time. This additional information allows you to determine the exact value of C.
Step 4: Obtain the position vs. time equation
The position vs. time equation can be obtained by rearranging the integrated equation:
s = 0.085t^2 - 0.43t + C,
where s represents the position and t represents time. The value of C will depend on the initial conditions or other information given in the problem.
To determine the position vs. time equation with accurate numerical constants, you need to know the value of the constant C or have additional information to calculate it precisely.