A particle moving in a striaght line with uniform deceleration has a velocity of 40m/s at point P. 20m/s at point Q and finally come at rest at point R where QR is equal to 50m. calculate the distance PQ. ii, time taken for PQ,. iii, time taken for PR.
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To solve this problem, we can use the equations of motion involving uniformly decelerating motion. The main equation we will be using is:
v^2 = u^2 + 2as
where:
v is the final velocity,
u is the initial velocity,
a is the acceleration (in this case, the deceleration, which is negative),
and s is the displacement.
Let's solve each part of the problem step by step:
i. To calculate the distance PQ:
We are given that the velocity at point P is 40 m/s, and at point Q, it is 20 m/s. The final velocity at point R is 0 m/s since the particle comes to rest there.
Using the equation v^2 = u^2 + 2as, we can find the displacement between P and Q:
(20)^2 = (40)^2 + 2a(s)
400 = 1600 + 2a(s)
2a(s) = -1200
We are also given that the displacement between Q and R (QR) is 50 m. So we can write:
QR = s = -50
Now we can solve for a:
-2400a = -1200
a = 0.5 m/s^2
To find the distance PQ, we substitute the values into the equation:
(20)^2 = (40)^2 + 2(0.5)(s)
400 = 1600 + s
s = -1200
The distance PQ is 1200 meters.
ii. To calculate the time taken for PQ:
We can use the equation v = u + at, where u is the initial velocity, a is the deceleration, and t is the time taken.
At point P, the particle has a velocity of 40 m/s. At point Q, the velocity is 20 m/s.
20 = 40 + (-0.5)t
20 - 40 = -0.5t
-20 = -0.5t
t = 40 seconds
Therefore, the time taken for PQ is 40 seconds.
iii. To calculate the time taken for PR:
We can use the equation v = u + at, where u is the initial velocity, a is the deceleration, and t is the time taken.
At point P, the particle has a velocity of 40 m/s. At point R, the velocity is 0 m/s.
0 = 40 + (-0.5)t
-40 = -0.5t
t = 80 seconds
Therefore, the time taken for PR is 80 seconds.
a = v^2/2x (use QR segment)
since a is constant use it for i and ii
eg. x = (vf^2 - vi^2)/2a