Question

For time t>0, a particle moves along the x-axis, and its position is given by the function s(t)=−t −4t⁴ +18t² +3. At what time does the particle obtain its maximum velocity?

A) t=0
B) t=1
C) t=2
D) t=3

Answer 1

The maximum velocity of a particle is reached when the acceleration is zero. By differentiating the given position function and setting the derivative (acceleration) equal to zero, we find that the maximum velocity occurs at t = 1 second.

Step-by-step explanation:

To find the time at which the particle obtains its maximum velocity, we first need to obtain the velocity function by differentiating the position function s(t) = -t - 4t⁴ + 18t² + 3 with respect to time t. Upon differentiating, we get the velocity function v(t) = ds/dt = -1 - 16t³ + 36t. The maximum velocity occurs when the derivative of the velocity function, which represents acceleration, is zero. So we must find the time t when a(t) = dv/dt = -48t² + 36 equals zero.

To find when a(t) is zero, solve the quadratic equation -48t² + 36 = 0, which gives us t = 1 and t = -1 (/t>/0 only). Since we are considering t > 0, we ignore t = -1. Therefore, the correct answer is B) t = 1. At t = 1 second, the particle reaches its maximum velocity.