Find the total distance traveled by the particle moving along a straight line with velocity v=sinpit for 0<t<2.
Distance travelled = integral of V dt
= Integral of sin (pi*t) dt
t = 0 to 2
= - cos(2) + cos(0) = 1 - cos(2) = 0.58385
To find the total distance traveled by the particle, we need to integrate the absolute value of the velocity function over the given time interval.
Given:
Velocity function: v = sin(pt), where p = 2π
Time interval: 0 < t < 2
To get the total distance traveled, follow these steps:
1. Integrate the absolute value of the velocity function over the given time interval:
∫|v| dt from 0 to 2
Since the velocity function v = sin(pt), the absolute value of v is |v|.
2. Determine the limits of integration:
The time interval is from t = 0 to t = 2. These values will be used as the limits of integration in the next step.
3. Integrate the absolute value of the velocity function:
∫|v| dt from 0 to 2
∫|sin(pt)| dt from 0 to 2
To calculate this integral, we need to consider the different cases when sin(pt) is positive or negative within the time interval.
a) When sin(pt) is positive:
In this case, the integral simply becomes:
∫sin(pt) dt from 0 to 2
The antiderivative of sin(pt) with respect to t is -1/p cos(pt):
= -1/p cos(pt)
Evaluate this expression at the upper (2) and lower (0) limits of integration:
= -1/p cos(p*2) - (-1/p cos(p*0))
= -1/p [cos(2p) - cos(0)]
= -1/p [cos(4π) - cos(0)] (since p = 2π)
= -1/p [1 - 1]
= 0
b) When sin(pt) is negative:
In this case, the integral becomes:
∫-sin(pt) dt from 0 to 2
The antiderivative of -sin(pt) with respect to t is 1/p cos(pt):
= 1/p cos(pt)
Evaluate this expression at the upper (2) and lower (0) limits of integration:
= 1/p cos(p*2) - 1/p cos(p*0)
= 1/p [cos(2p) - cos(0)]
= 1/p [cos(4π) - cos(0)] (since p = 2π)
= 1/p [1 - 1]
= 0
4. Add up both cases:
Since the result of both cases is zero, we can conclude that the total distance traveled by the particle is zero.
Therefore, the total distance traveled by the particle moving along a straight line with velocity v = sin(pt) for 0 < t < 2 is zero.