Given an equation v(t) that represents the velocity of an object (moving in a straight line) as a function of time, what is an equation s(t) that represents the object's position as a function of time?
Oh, position equations, my favorite! Alright, let's get this party started. If we have the velocity equation, v(t), we can find the position equation, s(t), by integrating the velocity equation with respect to time.
So, get your integration hats on and let's do this! The equation for position as a function of time, s(t), is the integral of the velocity equation, v(t), with respect to time:
s(t) = ∫v(t) dt
Now, go ahead and grab your favorite integration tool and get cracking. But don't worry, I won't make you integrate any equations involving clowns or banana peels. I'll save those for myself.
To find the equation s(t) that represents the object's position as a function of time, we need to integrate the velocity function v(t) with respect to time.
The position function s(t) is the integral of the velocity function v(t). In mathematical notation, we can express this as:
s(t) = ∫ v(t) dt
Integration allows us to find the antiderivative of v(t), representing the position function.
To integrate v(t), follow these steps:
1. Identify the expression for v(t). For example, if v(t) = 3t^2 + 5t, the expression is 3t^2 + 5t.
2. Add one to the exponent of each term. For the example above, we get: 3(t^2+1) + 5(t+1).
3. Divide each term by its new exponent. Continuing the example: (3/3)(t^2+1)^2 + (5/2)(t+1)^2.
4. Simplify each term. In the example, this would give us: t^3 + (5/2)t^2 + (3/2)t + constant.
Hence, the equation s(t) that represents the object's position as a function of time is:
s(t) = t^3 + (5/2)t^2 + (3/2)t + constant
where "constant" represents the constant of integration, which is determined by initial conditions or additional information provided.
To find the equation s(t) representing the position of the object as a function of time, we need to integrate the equation v(t) with respect to time. By doing so, we can determine the position function.
The first step is to identify the relationship between velocity and displacement. The velocity v(t) represents the rate of change of displacement with respect to time. In other words, it is the derivative of the position function s(t).
Therefore, if we integrate the velocity function v(t), it will undo the derivative and give us the position function s(t).
The equation for the position function s(t) can be found by integrating v(t) with respect to time:
s(t) = ∫ v(t) dt
The notation "∫" represents the integral sign, and dt indicates that we are integrating with respect to time.
Please provide the specific equation v(t) for further assistance in calculating s(t).