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.
To find the distance PQ, we need to use the equation of motion for uniformly decelerating motion, which is:
vf^2 = vi^2 + 2ad
Where:
- vf is the final velocity
- vi is the initial velocity
- a is the acceleration (which is negative in this case for deceleration)
- d is the distance traveled
Given:
- The particle has a velocity of 40 m/s at point P (which is the initial velocity, vi)
- The particle has a velocity of 20 m/s at point Q (which is the final velocity, vf)
- The distance QR is equal to 50 m
We can plug these values into the equation to find the distance PQ:
vf^2 = vi^2 + 2ad
(20 m/s)^2 = (40 m/s)^2 + 2 * a * PQ
400 m^2/s^2 = 1600 m^2/s^2 + 2 * a * PQ
2 * a * PQ = 1600 m^2/s^2 - 400 m^2/s^2
2 * a * PQ = 1200 m^2/s^2
a * PQ = 600 m^2/s^2
Since we know that the distance QR is 50 meters, we can set up another equation using the formula for uniformly decelerating motion:
vf^2 = vi^2 + 2ad
0 = (20 m/s)^2 + 2 * a * 50 m
0 = 400 m^2/s^2 + 100 m * a
a * 100 m = -400 m^2/s^2 (multiplying both sides by 100 m)
a = -4 m/s^2 (dividing both sides by 100 m)
Now, we can substitute the value of a into the previous equation:
-4 m/s^2 * PQ = 600 m^2/s^2
PQ = 600 m^2/s^2 / -4 m/s^2
PQ = -150 m
Since distance cannot be negative, we take the absolute value of PQ:
PQ = | -150 m | = 150 m
Therefore, the distance PQ is 150 meters.
ii. To calculate the time taken for PQ, we can use the equation of motion:
vf = vi + at
Where:
- vf is the final velocity (20 m/s)
- vi is the initial velocity (40 m/s)
- a is the acceleration (which is negative in this case for deceleration)
- t is the time
We need to solve for t:
20 m/s = 40 m/s + (-4 m/s^2) * t
20 m/s - 40 m/s = -4 m/s^2 * t
-20 m/s = -4 m/s^2 * t
t = -20 m/s / -4 m/s^2
t = 5 s
Therefore, the time taken for PQ is 5 seconds.
iii. To calculate the time taken for PR, we can use the equation of motion:
vf = vi + at
Where:
- vf is the final velocity (0 m/s)
- vi is the initial velocity (40 m/s)
- a is the acceleration (which is negative in this case for deceleration)
- t is the time
We need to solve for t:
0 m/s = 40 m/s + (-4 m/s^2) * t
40 m/s = -4 m/s^2 * t
t = 40 m/s / -4 m/s^2
t = -10 s
Therefore, the time taken for PR is 10 seconds.