Question

"What is the velocity function, v(t), for a particle moving along a line, given the interval 1 < t < 7, represented by the equation v(t) = t^2 - 4t - 12?"

Answer 1

The velocity function, represented by v(t) = t^2 - 4t - 12, is a quadratic equation that describes how the velocity of a particle changes over time in the interval 1 < t < 7.

Step-by-step explanation:

The velocity function v(t) for a particle moving along a line in the given interval 1 < t < 7 can be determined by the equation v(t) = t^2 - 4t - 12. To understand the motion of the particle, we can analyze this function, which represents how the velocity of the particle changes with time.

For the given equation, which is a quadratic function, we can find specific characteristics of the particle's motion, such as when the velocity is zero by finding the roots of the equation, the particle's maximum and minimum velocity within the interval, and how the velocity changes as time progresses.

If we want to explore further details like the particle's acceleration or position, we would need to take the derivative of the velocity function to find acceleration, or integrate the velocity function to find the position function.