Question

A particle that moves along a straight line has velocity v(t) = t²e⁻⁴ᵗ meters per second after t seconds. How many meters will it travel during the first t seconds (from time=0 to time=t)?

Answer 1

To find the meters traveled during the first t seconds, we need to find the integral of the velocity function using the power rule for integration and then evaluate the position function at time t.

Step-by-step explanation:

To determine the meters traveled during the first t seconds, we need to find the integral of the velocity function from time=0 to time=t. In this case, the velocity function is given as v(t) = t²e⁻⁴ᵗ.

To find the integral, we can use the power rule for integration. We have:^(t³e⁻⁴ᵗ)/(-4) + C, where C is the constant of integration.

Now, plugging in the limits of integration, we have:^(t³e⁻⁴ᵗ)/(-4) + C at time=t - (0³e⁻⁴(0))/(-4) - C at time=0

Simplifying this expression gives us the position function. Finally, we can plug in the value of t to find the number of meters traveled during the first t seconds.