A particle starts from rest and has an acceleration of 2m/s*2 for 10sec.After that, the particle travels for 30sec with constant speed and then undergoes a retardation of 4m/s*2 and comes back to rest. The total distance covered by the particle is?
x0 = 0
x1 = (1/2) 2 (100)
v1 = 2*10 = 20
x2 = x1 + 20 (30) = x1 + 600
v2 = x1 = 20
time retarding = t
0 = v2 - 4 t
so t = v2/4
then
x3 = x2+v2(v2/4)-(1/2)(4)(v2^2/16)
To find the total distance covered by the particle, we need to calculate the distance traveled during each phase of its motion and then add them together.
Phase 1: Acceleration for 10 seconds
Given:
Initial velocity (u) = 0 m/s (particle starts from rest)
Acceleration (a) = 2 m/s^2
Time (t) = 10 s
Using the kinematic equation:
distance (s) = ut + (1/2)at^2
s = (0)(10) + (1/2)(2)(10^2)
s = 0 + 10(10)
s = 100 m
Phase 2: Constant speed for 30 seconds
Given:
Time (t) = 30 s
Velocity remains constant since there is no acceleration.
Using the formula:
distance (s) = velocity (v) * time (t)
s = v*t
Since the velocity remains constant, s = v*t simplifies to:
s = constant velocity * 30
s = constant
Phase 3: Retardation (deceleration) until rest
Given:
Initial velocity (u) = constant velocity from phase 2
Acceleration (a) = -4 m/s^2 (negative because it is in the opposite direction to the constant velocity)
Final velocity (v) = 0 m/s (particle comes to rest)
Using the kinematic equation:
v^2 = u^2 + 2as
0 = u^2 + 2(-4)s
0 = u^2 - 8s
Since we know that s = constant from phase 2, we can substitute it into the equation:
0 = constant^2 - 8(constant)
0 = constant(constant - 8)
This equation has two possible solutions:
1. constant = 0 (particle remains at rest, no distance covered)
2. constant - 8 = 0 (particle transitions from constant speed to rest)
Since the question states that the particle comes to rest, we use constant - 8 = 0.
constant - 8 = 0
constant = 8
Phase 3 distance = constant = 8 m
Total distance covered = Phase 1 distance + Phase 2 distance + Phase 3 distance
Total distance covered = 100 m + constant + 8 m
Total distance covered = 100 m + 8 m
Total distance covered = 108 m
Therefore, the total distance covered by the particle is 108 meters.