A particle moves in a straight line with velocity given by v(t) =sin t metres per second. Find the total distance travelled by the particle in the first 4 seconds of motion
To find the total distance traveled by the particle in the first 4 seconds, we need to calculate the displacement for each second and then add them all together.
The displacement for each second can be calculated by integrating the velocity function over each second interval.
Let's calculate it step by step:
Step 1: Determine the interval for each second.
Since we want to find the distance traveled in the first 4 seconds, we have the following intervals:
- From 0 to 1 second
- From 1 to 2 seconds
- From 2 to 3 seconds
- From 3 to 4 seconds
Step 2: Calculate the displacement for each second interval.
For the first interval (0 to 1 second):
The displacement is given by the integral of the velocity function over the interval [0, 1].
∫[0, 1] sin t dt = [-cos t] from 0 to 1 = -cos(1) - (-cos(0)) = -cos(1) +1
For the second interval (1 to 2 seconds):
The displacement is given by the integral of the velocity function over the interval [1, 2].
∫[1, 2] sin t dt = [-cos t] from 1 to 2 = -cos(2) - (-cos(1)) = -cos(2) + cos(1)
For the third interval (2 to 3 seconds):
The displacement is given by the integral of the velocity function over the interval [2, 3].
∫[2, 3] sin t dt = [-cos t] from 2 to 3 = -cos(3) - (-cos(2)) = -cos(3) + cos(2)
For the fourth interval (3 to 4 seconds):
The displacement is given by the integral of the velocity function over the interval [3, 4].
∫[3, 4] sin t dt = [-cos t] from 3 to 4 = -cos(4) - (-cos(3)) = -cos(4) + cos(3)
Step 3: Add up the displacements for each interval to find the total distance traveled.
Total distance traveled = |-cos(1) + 1| + |-cos(2) + cos(1)| + |-cos(3) + cos(2)| + |-cos(4) + cos(3)|
Calculating the values:
Total distance traveled = 1.3817 + 1.1920 + 1.2206 + 1.0009 = 4.7946 meters
Therefore, the total distance traveled by the particle in the first 4 seconds is approximately 4.7946 meters.
To find the total distance traveled by the particle in the first 4 seconds, we need to integrate the velocity function over the interval [0,4].
The velocity function is given by v(t) = sin t.
To integrate this function, we can use the following indefinite integral:
∫ sin t dt = -cos t + C
where C is the constant of integration.
To find the definite integral over the interval [0,4], we substitute the upper and lower limits:
∫[0,4] sin t dt = [-cos(4) + C] - [-cos(0) + C]
Since the constant of integration cancels out, we have:
[-cos(4) + C] - [-cos(0) + C] = -cos(4) + cos(0)
Finally, we evaluate the cosine function at the respective angles:
Total distance = -cos(4) + cos(0) ≈ -0.6536 + 1
Therefore, the total distance traveled by the particle in the first 4 seconds is approximately 0.3464 meters.
integratiing we get
s(t) = -cost + c , where s(t) is distance and c is a constant
s(0( = -cos0 + c = -1+c
s(4) = -cos 4 + c
distance covered = -cos 4 + c - (-1+ c)
= -cos 4 + 1
= appr. 1.6536
make sure your calculator is set to RAD (radians)