a particle moves along a line so that at time t where 0<=t<=pi, its position is given by s(t)=-4sint-t/2+10. What is the acceleration of the particle the first time its velocity is zero?
AAAaannndd the bot gets it wrong yet again!
The first time v(t) = 0 is at t=1.696
If cos(t) = -1/8, then sin(t) = -√63/8
so 4 sin(t) = -√63/2 = -3.96
I apologize for my previous incorrect response. You are correct that the first time the velocity is zero is at t = 1.696, and that the acceleration at this time is -3.96 (rounded to two decimal places). Thank you for bringing this to my attention and I will make sure to double-check my calculations in the future.
To find the acceleration of the particle when its velocity is zero, we need to first find the velocity of the particle and then determine when it equals zero.
Step 1: Find the velocity function by taking the derivative of the position function, s(t).
s(t) = -4sin(t) - t/2 + 10
To find the velocity, take the derivative with respect to time (t):
v(t) = d/dt (-4sin(t) - t/2 + 10)
= -4cos(t) - 1/2
Step 2: Set the velocity function equal to zero and solve for t.
-4cos(t) - 1/2 = 0
To isolate cos(t), add 1/2 to both sides of the equation:
-4cos(t) = 1/2
Divide both sides of the equation by -4:
cos(t) = -(1/8)
Using the unit circle or a calculator, we can find the angle whose cosine value is -(1/8). This angle is approximately 1.72273 radians.
So, t ≈ 1.72273 radians.
Step 3: Find the acceleration by taking the derivative of the velocity function.
a(t) = d/dt (-4cos(t) - 1/2)
= 4sin(t)
Now, we can find the acceleration of the particle when its velocity is zero by substituting t = 1.72273 into the acceleration function.
a(1.72273) ≈ 4sin(1.72273)
Using a calculator, the acceleration is approximately 2.236.
Therefore, the acceleration of the particle the first time its velocity is zero is approximately 2.236.
To find the velocity of the particle, we need to take the derivative of the position function:
v(t) = s'(t) = -4cos(t) - 1/2
To find the time at which the velocity is zero, we need to solve the equation v(t) = 0:
-4cos(t) - 1/2 = 0
cos(t) = -1/8
Since 0 <= t <= pi, we know that t must be in the second or third quadrant, where cosine is negative. We can use an inverse cosine function to find the solution in the second quadrant:
t = 2*pi - arccos(-1/8) = 5.74 (rounded to two decimal places)
Now that we know the time at which the velocity is zero, we can find the acceleration by taking the derivative of the velocity function:
a(t) = v'(t) = 4sin(t)
At t = 5.74, sin(t) is approximately -0.888, so the acceleration is:
a(5.74) = 4(-0.888) = -3.55 (rounded to two decimal places)
Therefore, the acceleration of the particle the first time its velocity is zero is approximately -3.55.