A particle moving in a straight line was the uniform deceleration has a velocity of 40m/s at a point P,20m/s at a point Q and comes to rest at point R QR=50m, calculate distance PQ,time taken to cover PQ ,time taken to cover PR
To solve this problem, we can use the equations of motion. Let's break it down step by step:
1. Velocity at point P = 40 m/s
2. Velocity at point Q = 20 m/s
3. Distance between Q and R (QR) = 50 m
Now, we need to find the distance PQ, time taken to cover PQ, and time taken to cover PR.
Step 1: Finding the acceleration (a):
We know that the particle is uniformly decelerating. In uniform deceleration, acceleration (a) is negative.
Using the equation of motion: v^2 = u^2 + 2as
where v = final velocity, u = initial velocity, a = acceleration, and s = distance traveled
At point Q, the velocity is 20 m/s (v), and at point P, the velocity is 40 m/s (u). The distance traveled (s) between Q and P is unknown. The acceleration (a) is negative as it's a deceleration.
Using the equation, we get:
20^2 = 40^2 + 2 * a * s
400 = 1600 + 2as
2as = -1200
as = -600
Step 2: Finding the distance PQ:
We need to find the value of s, the distance between P and Q.
Using the equation of motion: v^2 = u^2 + 2as
where v = final velocity, u = initial velocity, a = acceleration, and s = distance traveled
At point Q, the velocity is 20 m/s (v), and at point P, the velocity is 40 m/s (u). The acceleration (a) is -600 (as determined in Step 1).
20^2 = 40^2 + 2 * (-600) * s
400 = 1600 - 1200s
-1200s = -1600
s = 4/3 = 1.33 m
Therefore, the distance PQ is approximately 1.33 meters.
Step 3: Finding the time taken to cover PQ:
We can use the equation of motion: v = u + at
where v = final velocity, u = initial velocity, a = acceleration, and t = time taken
At point Q, the velocity is 20 m/s (v), and at point P, the velocity is 40 m/s (u). The acceleration (a) is -600 (as determined in Step 1).
20 = 40 + (-600)t
-600t = -20
t = 1/30 = 0.033 s
Therefore, the time taken to cover PQ is approximately 0.033 seconds.
Step 4: Finding the time taken to cover PR:
To find the time taken to cover PR, we can consider the motion from point P to R.
Using the equation of motion: v^2 = u^2 + 2as
where v = final velocity, u = initial velocity, a = acceleration, and s = distance traveled
The particle comes to rest at point R, so the final velocity v is 0 m/s.
Using the acceleration value (-600) from Step 1, and the distance QR (50 m), we can solve for time (t).
0 = 40^2 + 2 * (-600) * 50
0 = 1600 - 60000
-60000 = -1600
t = -60000 / -1600
t = 37.5 s
Therefore, the time taken to cover PR is 37.5 seconds.
To summarize:
Distance PQ = 1.33 meters
Time taken to cover PQ = 0.033 seconds
Time taken to cover PR = 37.5 seconds