The acceleration of a particle is given by a(t)=cos(t)-sin(t). At t=0, the velocity is 1. During the time from pi < t< 2pi, what is the acceleration of the particle when the velocity is 0.
The less than signs are supposed to be less than or equal to signs.
Please help. Thank you.
Use <= for less than or equal.
a(t) = cos(t)-sin(t)
v(t) = sin(t)+cos(t) + c
v(0)=1, so c=0
v=0 when sin(t)+cos(t)=0.
That is, sin(t) = -cos(t)
sin(t)=cos(t) when t=π/4, so using that as a reference angle, that makes t = 3π/4 or 11π/4
Only 11π/4 is between π and 2π.
So, a(11π/4) = 1/√2 - (-1/√2) = √2
To determine the acceleration of the particle when the velocity is zero, we first need to find the time interval during which the velocity becomes zero.
Given that the acceleration of the particle is a(t) = cos(t) - sin(t), we can find the velocity function by integrating the acceleration function over time:
v(t) = ∫(a(t)) dt = ∫(cos(t) - sin(t)) dt
Integrating cos(t) gives us sin(t), and integrating -sin(t) gives us cos(t):
v(t) = sin(t) + cos(t)
Given that the velocity at t=0 is 1, we can substitute t=0 into v(t) to find the constant of integration:
v(0) = sin(0) + cos(0) = 0 + 1 = 1
So, the velocity function becomes:
v(t) = sin(t) + cos(t)
Next, we need to find the time interval within pi < t < 2π where the velocity becomes zero. This means we need to solve the equation v(t) = 0:
sin(t) + cos(t) = 0
To solve this equation, we can rewrite it as:
sin(t) = -cos(t)
Now, we can use the identity sin(t) = cos(t - π/2) to rewrite the equation as:
cos(t - π/2) = -cos(t)
Since cos(t1) = cos(t2) implies t1 = ±t2 + 2πn, where n is an integer, we have two cases to consider:
Case 1: t - π/2 = -t + 2πn
Simplifying the equation:
2t = π/2 + 2πn
t = π/4 + πn
Case 2: t - π/2 = t + 2πn
Simplifying the equation:
-π = 4πn
t = -π/4 + πn
Therefore, the time interval within π < t < 2π where the velocity becomes zero is:
-π/4 + πn < t < π/4 + πn
To find the acceleration at the moment the velocity is zero, we substitute these values of t into the given acceleration function a(t) = cos(t) - sin(t):
a(t) = cos(t) - sin(t)
So, the acceleration of the particle when the velocity is zero within the time interval π < t < 2π is given by the values of a(t) corresponding to the range of t mentioned above.