A uniformly accelerated particle covers three successive equal distances in time intervals t1, t2 and t3 respectively.
so whats your question ?
Show that 1/t1 - 1/t2 +1/t3 =3/t1+t2+t3
To determine the relationship between the time intervals and the distance covered by a uniformly accelerated particle, we can use the equations of motion.
Let's denote the distance covered in each time interval as d1, d2, and d3, and the corresponding time intervals as t1, t2, and t3, respectively. We want to find the relationship between these variables.
First, we need to recall the equation for the distance covered by a uniformly accelerated particle:
d = ut + (1/2)at^2
where d is the distance, u is the initial velocity, a is the acceleration, and t is the time.
Since we are dealing with a uniformly accelerated particle, the acceleration remains constant throughout. Therefore, we can write the equation for each time interval as follows:
For the first time interval:
d1 = ut1 + (1/2)at1^2
For the second time interval:
d2 = ut2 + (1/2)at2^2
For the third time interval:
d3 = ut3 + (1/2)at3^2
Next, we need to recognize that the initial velocity for each interval is the final velocity of the previous interval:
u2 = v1
u3 = v2
where v1 is the final velocity at the end of the first interval, and v2 is the final velocity at the end of the second interval.
We can also use the equations of motion for uniformly accelerated motion to relate the velocities and time intervals:
v = u + at
From the equation, we can express the final velocities as follows:
v1 = u1 + at1
v2 = u2 + at2
Now, let's use these equations to find the relationship between the time intervals and the distances covered.
Substituting the expressions for v1 and v2 into the equations for d1 and d2, we get:
d1 = (u1 + at1)t1 + (1/2)at1^2
d2 = (v1)t2 + (1/2)at2^2
Now, substituting the expression for v1 into the equation for d2, we get:
d2 = (u1 + at1)t2 + (1/2)at2^2
Expanding both equations, we have:
d1 = u1t1 + (1/2)at1^2
d2 = u1t2 + at1t2 + (1/2)at2^2
Finally, substituting the expressions for d1, d2, and d3 into the equations, we have:
u1t1 + (1/2)at1^2 = d1
u1t2 + at1t2 + (1/2)at2^2 = d2
u1t3 + at2t3 + (1/2)at3^2 = d3
These equations show the relationship between the time intervals and the distances covered by the uniformly accelerated particle. By solving these equations, you can find the specific values of the distances covered in each time interval.