To solve this problem, we can use the equations of motion for uniformly decelerating particles. Let's break down the problem step by step:
i. Calculating the distance PQ:
We know that the particle is decelerating, so the velocity at Q (20 m/s) is less than the velocity at P (40 m/s). We also know that the particle comes to rest at point R. Let's use the equation:
v² = u² + 2as
where:
v = final velocity
u = initial velocity
a = acceleration
s = distance
For PQ, the initial velocity (u) is 40 m/s, the final velocity (v) is 20 m/s, and the acceleration (a) is uniform. We need to determine the distance (s) for PQ.
Plugging in the values, we get:
20² = 40² + 2a * s
400 = 1600 + 2a * s
2a * s = -1200
Since the acceleration is negative (deceleration), we can write -a instead of a. Therefore:
-2a * s = 1200
Since QR is equal to 50 m, we can say s + 50 = s. Substituting this into the equation:
-2a * (s + 50) = 1200
Simplifying:
-2a * s - 100a = 1200
The question doesn't provide the value of the acceleration, so we can't find the exact value of s. However, we can set up an algebraic expression:
-2a * s - 100a = 1200
ii. Calculating the time taken for PQ:
To calculate the time taken for PQ, we need to know either the acceleration or the distance. As mentioned earlier, the question doesn't provide the acceleration value. Therefore, we can't determine the exact time. However, we can provide an algebraic expression:
Let t be the time taken for PQ.
Using the equation of motion:
v = u + at
Substituting the values:
20 = 40 + a * t
Simplifying:
a * t = -20
Again, since we don't know the acceleration value, we can't compute the exact time. However, we now have an expression: a * t = -20.
iii. Calculating the time taken for PR:
Since we know the distance QR is equal to 50 m, we can use the equation of motion to calculate the time taken for PR. The velocity at point R is 0 m/s because the particle comes to rest there.
v = u + at
0 = 40 + a * t
Simplifying:
a * t = -40
Similar to the previous cases, we can't determine the exact time without knowing the acceleration value. However, we have the expression: a * t = -40.
In summary:
i. The distance PQ can be expressed as -2a * s - 100a = 1200.
ii. The time taken for PQ can be expressed as a * t = -20.
iii. The time taken for PR can be expressed as a * t = -40.
Please note that without knowing the value of acceleration (a), we can't find the exact values for the distance PQ or the times taken for PQ and PR.