Particle Motion from Equation

Question

A particle moves along the x-axis so that at time t≥0 its position is given by x(t)=t^5−5t^4. Determine all intervals when the acceleration of the particle is negative.

Answer 1

From the problem we are given the equation for the position of the particle. We need to find the equation for its acceleration, since they are asking for whenever the acceleration is negative. If you recall, we can take the derivative of the position, which gives us its velocity, and then we can take the derivative of the velocity to get its acceleration.

x(t) = t⁵ - 5t⁴ (position equation)

x(t)' = 5t⁴ - 20t³ (first derivative - velocity)

x(t)'' = 20t³ - 60t² (second derivative - acceleration)

Now the last thing we need is to set up the equation for acceleration as an inequality to find out when it is negative:

1. 20t³ - 60t² < 0
2. 20t²(t-3) < 0 (factor out the t²)
3. 20t² < 0 and t - 3 < 0

solving for t, we get t < 3, which is when acceleration is negative. The interval is [0, 3).

Answer 2

x = t^5 -5t^4

x' = velocity = 5t^4 -20t^3

x" =acceleration = 20t^3 -60t^2 < 0

20t^2(t-3) < 0

t<3

from t=0 to t< 3 seconds, acceleration < 0

t is in the interval [0,3)