4. (a) If the position of a particle is given by x = 20+ 5t^3,where x is in meters and t is in seconds, when if ever is the particle's velocity zero? (b) When is its acceleration a zero? (c) When is a negative? Positive? (d) Graph x(t), v(t), and a(t).
Answers
Answer:
The derivative of a position function will give us the velocity and the second derivative of a position function, or the first derivative of a velocity function will give us the acceleration function.
Thus, let us first take the derivative of the position function.
v(t)=\frac{dx(t)}{dt}=-15t^{2}+20v(t)=
dt
dx(t)
=−15t
2
+20
Let us now take the derivative of the velocity function we just obtained.
a(t)=\frac{dv(t)}{dt}=-30ta(t)=
dt
dv(t)
=−30t
From these equations, we will now answer the questions.
a) To figure out when the velocity of the particle is zero, we simply find tt that gives us 0 as it’s answer. Therefore:
0=-15t^{2}+200=−15t
2
+20
t=\sqrt{\frac{20}{15}}t=
15
20
t=1.2 st=1.2s
b) As before, to find when the acceleration is zero, we need to find the value of tt that gives us 0 as it’s answer in our function. Therefore:
0=-30t0=−30t
t=0t=0
c) We can see that because our acceleration, a=-30ta=−30t, for any value of t>0, we would get a negative value.
d) We can also see that for any value of t<0, we would get a positive acceleration value.