Question

For t≥0, the position of a particle moving along the x-axis is given by x(t)=sint−cost . What is the acceleration of the particle at the point where the velocity is first equal to 0 ?

Answer 1

The acceleration of a particle moving along the x-axis with a position function x(t) = sin(t) - cos(t) is 0 m/s² when the velocity is first equal to zero, which occurs at t = π/4 radians.

Step-by-step explanation:

When a particle's position along the x-axis is given by x(t) = sin(t) - cos(t), we can determine the particle's acceleration at the point where the velocity is first equal to zero by differentiating the position function with respect to time to get the velocity, and then again to get the acceleration.

The velocity function v(t) is the first derivative of x(t) with respect to time, so v(t) = cos(t) + sin(t). The acceleration function a(t) is the derivative of the velocity v(t), so a(t) = -sin(t) + cos(t).

To find when the velocity is first equal to zero, we set v(t) = 0, which gives us the equation cos(t) + sin(t) = 0. By solving this equation, we find that the velocity is first zero at t = π/4 radians. Then we evaluate the acceleration function at t = π/4 to find the acceleration at this point, which will be a(π/4) = -sin(π/4) + cos(π/4). Calculation of this expression yields a(π/4) = √2/2 - √2/2 = 0 m/s².

Therefore, the acceleration of the particle at the point where the velocity is first equal to zero is 0 m/s².