Particle motion At time t ≥ 0, the velocity of a body moving along the horizontal s-axis is v = t² − 4t + 3.


b. When is the body moving forward? Backward?

Verified step by step guidance

1. To determine when the body is moving forward or backward, we need to analyze the velocity function v(t) = t² − 4t + 3.
2. The body is moving forward when the velocity v(t) is greater than zero, and moving backward when v(t) is less than zero.
3. First, find the critical points by setting the velocity function equal to zero: t² − 4t + 3 = 0. Solve this quadratic equation to find the values of t where the velocity changes sign.
4. Use the quadratic formula t = (-b ± √(b² - 4ac)) / 2a, where a = 1, b = -4, and c = 3, to find the roots of the equation.
5. Once the critical points are found, test intervals around these points to determine the sign of v(t) in each interval. This will tell you when the body is moving forward (v(t) > 0) and when it is moving backward (v(t) < 0).

Key Concepts

Velocity and Direction

Velocity indicates the speed and direction of a particle's motion along a path. A positive velocity means the particle is moving forward, while a negative velocity indicates backward motion. To determine when the body moves forward or backward, analyze the sign of the velocity function over time.

Quadratic Functions

The velocity function v = t² − 4t + 3 is a quadratic function, which is characterized by its parabolic graph. The roots of the quadratic equation, found using the quadratic formula, indicate the points where the velocity changes sign, helping to identify intervals of forward and backward motion.

Solving Inequalities

To determine when the body moves forward or backward, solve the inequality v(t) > 0 for forward motion and v(t) < 0 for backward motion. This involves finding the roots of the quadratic equation and testing intervals between these roots to see where the inequality holds true.