A car is stopped at a red light, and starts moving at time t = 0 when the light turns green, with an acceleration given by a(t) = √t, with t given in seconds, and distances given in metres.
a. Find a Formul
s = 1/2 at^2 = 1/2 t^(5/2)
To find a formula for the position of the car, we need to integrate the acceleration function.
The formula for acceleration is given as a(t) = √t.
First, we integrate the acceleration function to get the velocity function:
v(t) = ∫ a(t) dt = ∫ √t dt.
To integrate the function √t, we can use the power rule of integration. The power rule states that if we have an integral of the form ∫ x^n dx, then the result is (1/(n+1)) * x^(n+1).
Applying this rule to the integral of √t, we have:
v(t) = ∫ √t dt = (2/3) * t^(3/2) + C1,
where C1 is the constant of integration.
Next, we integrate the velocity function to find the position function:
s(t) = ∫ v(t) dt = ∫ [(2/3) * t^(3/2) + C1] dt.
Again, using the power rule of integration, we have:
s(t) = (2/3) * (2/5) * t^(5/2) + C1 * t + C2,
where C2 is another constant of integration.
So, the formula for the position of the car as a function of time is:
s(t) = (4/15) * t^(5/2) + C1 * t + C2.
Now, if we have specific initial conditions (position and velocity at t=0), we can determine the values of C1 and C2 to get a more precise formula.