A uniformly accelerated particle covers three successive equal distances in time intervals t1, t2 and t3 respectively.
To solve this problem, we can use the equations of motion for uniformly accelerated particle. Let's assume the initial velocity of the particle is u and the final velocity is v. The acceleration of the particle is denoted by a.
The equations of motion are:
1. v = u + at
2. s = ut + (1/2)at^2
3. v^2 = u^2 + 2as
Given that the particle covers three equal distances, let's call this distance as 's'.
First, we need to find the time intervals in terms of t1, t2, and t3. Let's assume the time taken to cover the first distance 's' is t1, the time taken to cover the second distance 's' is t2, and the time taken to cover the third distance 's' is t3.
Using equation 2, we have:
s = ut1 + (1/2)at1^2
s = ut2 + (1/2)at2^2
s = ut3 + (1/2)at3^2
Next, we can find the final velocity of the particle at the end of each interval using equation 1:
v1 = u + at1
v2 = u + at2
v3 = u + at3
Now, using equation 3, we can relate the square of the final velocity v with the initial velocity u, acceleration a, and distance s:
v1^2 = u^2 + 2as
v2^2 = u^2 + 2as
v3^2 = u^2 + 2as
Since the distances and time intervals are equal, we can assume that the initial and top velocities are also constant throughout the motion (meaning u = v1 = v2 = v3).
With these equations, we can solve for u, a, and s. Substitute u = v1 = v2 = v3 into equation 3 and solve for a:
a = (v3^2 - v1^2) / (2s)
Once we find the acceleration, we can substitute it into any of the equations (1 or 2) to solve for a specific value.
Remember to use consistent units for time, distance, and velocity when plugging in values into the equations.