a race car driver on a straight track starts the tryout from rest at t = 0 and x = 0. the car accelerates with an acceleration given by a(t) = 0.25t (m/s^2). how far eill the car travel in 12 secs?
To find how far the car will travel in 12 seconds, we first need to determine the car's velocity as a function of time. We can do this by integrating the acceleration function.
The acceleration function is given as a(t) = 0.25t (m/s^2).
To find the velocity function v(t), we integrate the acceleration function with respect to time:
∫ a(t) dt = ∫ 0.25t dt
This integration gives us:
v(t) = 0.25 * (t^2 / 2) + C
Applying the initial condition v(0) = 0 (since the car starts from rest), we can determine the constant C:
v(0) = 0.25 * (0^2 / 2) + C
0 = 0 + C
C = 0
Therefore, the velocity function is:
v(t) = 0.25 * (t^2 / 2)
Now, to find the distance traveled by the car in 12 seconds, we need to integrate the velocity function over the time interval [0, 12]:
∫ v(t) dt = ∫ [0, 12] 0.25 * (t^2 / 2) dt
Evaluating the definite integral, we find:
∫ [0, 12] 0.25 * (t^2 / 2) dt = 0.25 * [t^3 / 6] | from 0 to 12
Substituting the limits of integration, we get:
0.25 * [(12^3 / 6) - (0^3 / 6)]
= 0.25 * [1728 / 6]
= 0.25 * 288
= 72
Therefore, the car will travel a distance of 72 meters in 12 seconds.