a particle starts from rest and travels a distance x with uniform acceleration, then it travels a distance 2x with uniform speed ,finally it travels a distance of 3x with uniform retardation and comes to rest. If the whole motion of the particle is a straight line, then the ratio of its average velocity to maximum velocity is?
To find the ratio of the average velocity to the maximum velocity, we need to determine the values of average velocity and maximum velocity separately.
Let's break down the problem into three parts:
1. Particle travels a distance x with uniform acceleration:
In this part, the particle starts from rest and travels a distance x with uniform acceleration. Let's denote the time taken for this part as t1.
We can use one of the kinematic equations to relate distance, acceleration, and time:
x = 0.5 * a * t1^2
Since the particle starts from rest, its initial velocity (u) is 0. Using another kinematic equation relating velocity, acceleration, and time:
v = u + a * t1
v = 0 + a * t1
v = a * t1
Now, we can solve these two equations simultaneously to find the value of acceleration (a) and time (t1).
2. Particle travels a distance 2x with uniform speed:
In this part, the particle travels a distance 2x with uniform speed. Let's denote the time taken for this part as t2.
Since the speed is uniform, the acceleration is zero. Therefore, the velocity remains constant during this part. Let's denote this constant velocity as v2.
The distance covered (2x) is given by:
2x = v2 * t2
We can solve this equation to find the value of the constant velocity v2 and the time t2.
3. Particle travels a distance 3x with uniform retardation:
In this part, the particle travels a distance 3x with uniform retardation. Let's denote the time taken for this part as t3.
The retardation is in opposition to the motion, so we take it as a negative value. Denoting the retardation as -a, we can use the kinematic equation for uniformly accelerated motion:
3x = v2 * t3 + 0.5 * (-a) * t3^2
Now, we can solve this equation to find the value of time t3.
Once we have the values of t1, t2, and t3, we can calculate the average velocity (Vavg) and maximum velocity (Vmax).
The average velocity is given by:
Vavg = Total Distance / Total Time
Vavg = (x + 2x + 3x) / (t1 + t2 + t3)
Vavg = 6x / (t1 + t2 + t3)
The maximum velocity is given by the value of velocity (v) obtained in the first part of the motion (when the particle is under uniform acceleration):
Vmax = v = a * t1
Finally, the ratio of average velocity to maximum velocity is:
Ratio = Vavg / Vmax
Ratio = (6x / (t1 + t2 + t3)) / (a * t1)
So, to find the ratio, we need to determine the values of a, t1, t2, and t3 using the equations provided and then substitute those values in the ratio formula.