Rectilinear Motion with Variable Acceleration - Problem 9 - Kinematics of Particles

TL;DR

This analysis breaks down a problem involving variable acceleration and provides step-by-step calculations to find the maximum velocity and position of a particle at a specific time.

Transcript

hi friends it's all problem on variable acceleration just see what is given in problem acceleration of a particle is defined by relation a equal to 100 sine of pi t divided by 2 millimeter per second square that means unit of acceleration is millimeter per second square that means when you will find out maybe velocity or position that is in millime... Read More

Key Insights

• 😑 The given problem involves a particle with variable acceleration, expressed as a = 100sin(πt/2) mm/s^2.
• 🥰 The equation for velocity is obtained by integrating the acceleration equation, resulting in v = -200π^(-1)cos(πt/2) + 200π^(-1).
• ☺️ The position of the particle is determined by integrating the equation for velocity, leading to x = -400π^(-1)sin(πt/2) + 200t.
• 😃 The maximum velocity occurs when the acceleration is zero, which is found at t = 2 seconds.
• 😀 The maximum velocity is calculated to be 400/π m/s.
• 💬 The position of the particle at t = 4 seconds is determined to be 254.65 mm.

Questions & Answers

Q: How is the equation for velocity derived from the given equation for acceleration?

The equation for velocity is obtained by integrating the equation for acceleration with respect to time. By simplifying the integral, we arrive at v = -200π^(-1)cos(πt/2) + 200π^(-1).

Q: What condition is necessary for the velocity to be at its maximum?

For the velocity to be at its maximum, the derivative of velocity with respect to time (dv/dt) must be equal to zero. In this case, dv/dt represents the acceleration. Therefore, we set the equation for acceleration equal to zero and solve for the time t.

Q: How is the maximum velocity calculated using the equation for velocity?

To find the maximum velocity, we substitute the time value obtained from setting the acceleration equation equal to zero into the equation for velocity. By plugging t = 2 seconds into the equation v = -200π^(-1)cos(πt/2) + 200π^(-1), we can calculate the maximum velocity, which is 400/π m/s.

Q: How is the position of the particle determined at t = 4 seconds?

The position of the particle at t = 4 seconds is found by substituting t = 4 into the equation for position (x = -400π^(-1)sin(πt/2) + 200t). By evaluating this equation, we can calculate the position as 254.65 mm.

Summary & Key Takeaways

• The problem involves a particle with variable acceleration described by the equation a = 100sin(πt/2) mm/s^2. The goal is to find the maximum velocity and position of the particle at t = 4 seconds.

• By integrating the acceleration equation with respect to time, the equation for velocity (v) is obtained: v = -200π^(-1)cos(πt/2) + 200π^(-1).

• The equation for position (x) is then found by integrating the velocity equation, resulting in x = -400π^(-1)sin(πt/2) + 200t.