if a particle moves along a line so that at time t where t [0,pi] its position is given by s(t)=-4cost-t^2/2+10 what is the velocity of the particle when its acceleration is zero?
so i found
v(t) = 4sint-t
a(t)=4cost-1 = 0 solved for t plugged it into v(t) and i got the wrong answer i got 3.2 something but the choices are
-5.19
.74
1.32
2.55
8.13
please help thank you
Did you have your calculator set to radians ?
I get
t = arcsin(1/4)
= 1.318116
then
v(1.318116) = 4sin(1.318116) - 1.318116
= 2.55486
close to one of the choices,
(they should have rounded to 2.56)
3.969
Ah, I see you're having trouble with the velocity of the particle when its acceleration is zero. Well, let's see if I can put a smile on your face and help you out!
First, you correctly found the expression for velocity: v(t) = 4sint - t.
To find when the acceleration is zero, you set the expression for acceleration equal to zero and solve:
a(t) = 4cost - 1 = 0
Now, let's solve this equation to find the value of t:
4cost = 1
cost = 1/4
Now, let me just tune up my funny bone dial... Ah, yes! It seems that at t = π/3 and t = 5π/3, the cosine of an angle is equal to 1/4.
Now, let's plug these values of t into the expression for velocity, v(t), to find the velocity when the acceleration is zero:
v(π/3) = 4sin(π/3) - π/3 = . . . Wait for it! Drumroll, please . . .
-0.51 (approximately)
v(5π/3) = 4sin(5π/3) - 5π/3 = . . . And here it comes . . .
2.51 (approximately)
Oh, darn. Looks like my calculations didn't match any of the choices you provided. Maybe you should double-check your work just to be sure. But hey, at least we had a good laugh along the way, right? Keep those funny bones tickling!
To find the velocity of the particle when its acceleration is zero, we first need to find the values of t where the acceleration is zero.
Given that the acceleration of the particle is given by a(t) = 4cost - 1, we need to solve the equation a(t) = 0 for t.
4cost - 1 = 0
Simplifying: 4cost = 1
Dividing both sides by 4: cost = 1/4
To find the values of t where this equation is true, we need to find the inverse cosine (arc cosine) of both sides:
t = arccos(1/4)
Since t is in the range [0, π] as given in the question, we only need to find the appropriate solution within that range.
Using a calculator, the value of arccos(1/4) is approximately 1.3181 (rounded to four decimal places).
Now that we have the value of t where the acceleration is zero, we can substitute this value into the equation for velocity, v(t) = 4sint - t.
v(t) = 4sin(1.3181) - 1.3181
Evaluating this expression, we get approximately 2.5548 (rounded to four decimal places).
Therefore, the velocity of the particle when its acceleration is zero is approximately 2.5548.
None of the given answer choices match this result exactly. It is possible there was an error in the calculation or presentation of the answer choices.