Question

The velocity of a moving object in meters per second is given by v(t) = -2t + 4, where is the time in seconds. What is the displacement of the object from t=1 second to t = 4 seconds?

Answer 1

To find the displacement of the object from t = 1 second to t = 4 seconds, we need to calculate the integral of the velocity function v(t) over the interval [1, 4]. The displacement is given by the definite integral:

displacement = ∫[1,4] v(t) dt

First, let's find the antiderivative of v(t):

∫ v(t) dt = ∫ (-2t + 4) dt

Using the power rule of integration, we get:

= -t^2 + 4t + C

Now, we can evaluate the definite integral over the interval [1, 4]:

displacement = [-t^2 + 4t]₁̶=[1,4] = (-4^2 + 4(4)) - (-1^2 + 4(1)) = (-16 + 16) - (-1 + 4) = 0 - (-3) = 3

Therefore, the displacement of the object from t = 1 second to t = 4 seconds is 3 meters.