Motion of a Particle?

📌 CONCEPT: The position of a particle at time t can be found by integrating its velocity v(t) with respect to time, assuming the particle starts from rest at the origin.

📐 RULE / FORMULA: The position of the particle is given by the definite integral of its velocity function, i.e., s(t) = ∫v(t)dt, where s(t) is the position at time t.

💡 WORKED EXAMPLE: To find the position of the particle at time t = 3 seconds, we first integrate the given velocity function v(t) = 2t - 5 with respect to time: s(t) = ∫(2t - 5)dt = t^2 - 5t + C. Since the particle starts from rest at the origin, we have s(0) = 0, which gives C = 0. Therefore, the position function is s(t) = t^2 - 5t. Now, we evaluate this function at t = 3 to get the position at time t = 3 seconds: s(3) = 3^2 - 5(3) = 9 - 15 = -6.

⚠️ COMMON MISTAKE: Students often forget to evaluate the constant of integration (C) using the initial condition s(0) = 0, which leads to incorrect position functions.