the velocity in m/sec of a particle moving along the x-axis is given by the function v(t) = 7cos3t, 0 < t < pi/2
Find the particle's displacement for the given time interval
Displacement is the ending point minus the starting point.
Ending point is at t=PI/2
To find the displacement of the particle, we need to calculate its position at the starting point and at the ending point, and then subtract the starting position from the ending position.
Given that the velocity function of the particle is v(t) = 7cos(3t), we need to find its position function p(t) to determine the starting and ending positions.
The position function can be found by integrating the velocity function. The integral of cos(3t) is sin(3t)/3. So, the position function is:
p(t) = (7 sin(3t))/3 + C
To find the constant C, we need to consider the initial condition. Since the starting point is not given, we will assume it to be p(0) = 0. Plugging in t = 0 into the position function, we get:
0 = (7 sin(0))/3 + C
C = 0
Therefore, the position function is:
p(t) = (7 sin(3t))/3
Now, we can find the positions at the starting and ending points:
Starting position: p(0) = (7 sin(0))/3 = 0
Ending position: p(π/2) = (7 sin(3π/2))/3 = (7 * (-1))/3 = -7/3
Finally, we can calculate the displacement by subtracting the starting position from the ending position:
Displacement = Ending position - Starting position
= -7/3 - 0
= -7/3
Therefore, the particle's displacement for the given time interval is -7/3 units.