A particle moves on the x-axis so that its velocity at any time t is given by v(t) = sin 2t. At t = 0, the particle is at the origin.
a)For 0 ≤ t ≤ π, find all values of t for which the particle is moving to the left.
b)Write an expression for the position of the particle at any time t.
c)For 0 ≤ t ≤ π/2, find the average value of the position function determined in part b.
a)For 0 ≤ t ≤ π, the particle is moving to the left when sin 2t < 0. This occurs when t = π/2, 3π/2, 5π/2, etc.
b)The position of the particle at any time t is given by the integral of the velocity function, which is x(t) = -1/2 cos 2t.
c)For 0 ≤ t ≤ π/2, the average value of the position function is given by the integral of x(t) from 0 to π/2, divided by π/2. This is equal to -1/4.
a) To find the values of t for which the particle is moving to the left, we need to determine when the velocity function v(t) is negative.
Given v(t) = sin 2t, we know that the sine function is negative in the second and third quadrants. Therefore, we need to find the values of t in the interval 0 ≤ t ≤ π where 2t falls in the second or third quadrant.
In the second quadrant, 90° < 2t < 180°, which can be written as π/2 < 2t < π. To solve for t, we divide each side of the inequality by 2: π/4 < t < π/2.
In the third quadrant, 180° < 2t < 270°, which can be written as π < 2t < 3π/2. Dividing each side by 2 gives us π/2 < t < 3π/4.
Therefore, for 0 ≤ t ≤ π, the particle is moving to the left when π/4 < t < π/2 or π/2 < t < 3π/4.
b) To find the position of the particle at any time t, we need to integrate the velocity function v(t) with respect to t.
∫ v(t) dt = ∫ sin 2t dt
Using the identity ∫ sin x dx = -cos x + C, where C is the constant of integration, we have:
∫ sin 2t dt = -1/2 cos 2t + C
Since the particle is at the origin at t = 0, we can solve for the constant of integration, C, by substituting t = 0 into the equation:
0 = -1/2 cos 0 + C
0 = -1/2 + C
C = 1/2
Therefore, the expression for the position of the particle at any time t is:
x(t) = -1/2 cos 2t + 1/2
c) To find the average value of the position function x(t) for 0 ≤ t ≤ π/2, we need to calculate the definite integral of x(t) over this interval and divide it by the length of the interval.
∫[0,π/2] x(t) dt = ∫[0,π/2] (-1/2 cos 2t + 1/2) dt
= -1/4 sin 2t + 1/2t |[0,π/2]
= (-1/4 sin π + 1/2π/2) - (-1/4 sin 0 + 1/2 * 0)
= (-1/4 * 0 + 1/2π/2) - (0 + 0)
= 1/π - 0
= 1/π
Therefore, the average value of the position function x(t) for 0 ≤ t ≤ π/2 is 1/π.
a) To find the values of t for which the particle is moving to the left, we need to consider the sign of the velocity function v(t) = sin 2t.
The velocity function is positive when sin 2t > 0, which occurs when 0 < 2t < π/2 or π < 2t < 3π/2. This means that the particle is moving to the right for 0 < t < π/4 and 3π/4 < t < π.
The velocity function is negative when sin 2t < 0, which occurs when π/2 < 2t < π or 3π/2 < 2t < 2π. This means that the particle is moving to the left for π/4 < t < π/2 and π < t < 3π/4.
Therefore, the values of t for which the particle is moving to the left in the interval 0 ≤ t ≤ π are π/4 < t < π/2 and π < t < 3π/4.
b) To find the expression for the position of the particle at any time t, we need to integrate the velocity function v(t) = sin 2t with respect to t.
∫v(t) dt = ∫sin 2t dt
= -1/2 cos 2t + C
Since the particle is at the origin at t = 0, we can find the value of C by substituting t = 0 into the position function. Since cos 0 = 1, we have:
-1/2 cos 2(0) + C = 0
-1/2(1) + C = 0
-1/2 + C = 0
C = 1/2
Therefore, the expression for the position function is:
x(t) = -1/2 cos 2t + 1/2
c) To find the average value of the position function x(t) for 0 ≤ t ≤ π/2, we need to evaluate the definite integral:
1/(π/2 - 0) ∫[0, π/2] (-1/2 cos 2t + 1/2) dt
Simplifying the integral, we get:
1/(π/2) ∫[0, π/2] (-1/2 cos 2t + 1/2) dt
2/π ∫[0, π/2] (-1/2 cos 2t + 1/2) dt
2/π (-1/4 sin 2t + 1/2t) evaluated from 0 to π/2
Plugging in the limits of integration, we have:
2/π (-1/4 sin 2(π/2) + 1/2(π/2)) - 2/π (-1/4 sin 2(0) + 1/2(0))
2/π (-1/4 sin π + π/4)
Using the fact that sin π = 0, we can simplify further:
2/π (0 + π/4)
= 2/π * π/4
= 1/2
Therefore, the average value of the position function x(t) for 0 ≤ t ≤ π/2 is 1/2.