Find the position s as a function of time t from the given velocity v=ds/dv. Then evaluate the constant of integration so that s = s0 when, t = 0. 1. v = 3t2 , s0 = 4 I don't quite understand what the question is asking or what is s0 mean. Than
v = ds/dt or v = ds/dv ?? Please do not confuse me.
if ds/dt = v = 3 t^2
then
s = t^3 + c
when t = 0, s = s0 = 4 so constant of integration c = 4
s = t^3 + 4
In this problem, you are given the velocity v as a function of time t, which is v = 3t^2.
To find the position s as a function of time t, you need to integrate the velocity function with respect to t. The integral of v(t) with respect to t will give you s(t):
s(t) = ∫ v(t) dt
Substituting the given velocity function, we have:
s(t) = ∫ 3t^2 dt
To evaluate the constant of integration, we need to know the initial position s0. In this case, s0 is given as 4 units.
To find the position function, we integrate the velocity function:
s(t) = ∫ 3t^2 dt
= t^3 + C
Now, we can solve for the constant of integration C by using the initial condition s(t=0) = s0:
s(0) = (0)^3 + C
= C
Since s(0) = 4, we can substitute this value into the equation above:
C = 4
Therefore, the position function s(t) is:
s(t) = t^3 + 4
The question is asking you to find the position, denoted as "s," as a function of time, denoted as "t," given the velocity function v = ds/dt. In other words, you need to find an equation that expresses the position of an object as a function of time based on its velocity.
To find the position function, you need to integrate the velocity function with respect to time. Let's do that step by step:
Given velocity function: v = 3t^2
To find the position function, we integrate the velocity function with respect to time:
∫v dt = ∫(3t^2) dt
Integrating, we get:
s = ∫(3t^2) dt
To integrate 3t^2, we use the power rule of integration:
s = t^3 + C
Here, C is the constant of integration. It arises because when we integrate, there is usually an unknown constant that can take any value.
Now, to evaluate the constant of integration, we use the given condition that s = s0 when t = 0. In other words, when time t is equal to 0, the position s is equal to a specific value denoted as s0.
Given condition: s0 = 4, t = 0
Plugging these values into the position function, we have:
s0 = (0)^3 + C
s0 = 0 + C
s0 = C
Therefore, the constant of integration, C, is equal to s0, which is given as 4.
Finally, plugging the value of C back into the position function, we have:
s = t^3 + C
s = t^3 + 4
So, the position function as a function of time is s = t^3 + 4.