In a test run, a new train travels along a straight-line track. Data obtained from the speedometer indicate that the velocity of the train at any time t can be described by the velocity function
v(t) = 8t (0 ¡Ü t ¡Ü 30)
a. Find the position function of the train.
b. Find the position after 3 seconds. (Note: the train starts from the beginning of the track so when t = 0, the integration constant, C = 0.)
see your latest post.
To find the position function of the train, we need to integrate the velocity function with respect to time.
a. The velocity function is given as v(t) = 8t. Integrating this function will give us the position function.
To integrate v(t), we can use the power rule of integration, which states that the integral of t^n with respect to t is (1/(n+1)) * t^(n+1). In this case, we have v(t) = 8t, so n = 1.
∫v(t) dt = ∫(8t) dt = (8/2) * t^2 + C = 4t^2 + C
Since C is the integration constant and we know that the train starts from the beginning of the track (when t = 0), we can set C = 0.
So, the position function of the train is given by p(t) = 4t^2.
b. To find the position after 3 seconds, we can substitute t = 3 into the position function.
p(3) = 4(3)^2 = 36 units
Therefore, the position of the train after 3 seconds is 36 units.