P, Q and R start from the same place X at (a) kmph, (a+b) kmph and (a+2b) kmph respectively.
If Q starts p hours after P, how many hours after Q should R start, so that both Q and R overtake P at the same time?
t1+p = t2
so:
(a+b)/t2 = (a+2b)/(t2+h)
h= ((a+2b)*t2)/(a+b) - t2
h=b*t2/(a+b)
To solve this problem, we need to determine the time it takes for both Q and R to overtake P.
Let's assume that both Q and R overtake P after t hours.
During that time, Q will cover a distance of t * (a+b) km, and P will cover a distance of t * a km since Q started p hours after P.
Now, we have the equation: t * (a+b) = t * a
Simplifying the equation, we get:
t * a + t * b = t * a
b * t = 0
Since t cannot be equal to zero (as all three starting times are positive), we can conclude that b must be equal to zero.
Therefore, Q and R should start at the same time to overtake P at the same time.
To solve this problem, we need to use the concept of relative speed and distances. Let's assume that the time taken by P, Q, and R to overtake each other is t hours.
We know that the distance covered by an object is given by speed multiplied by time. Therefore, the distance covered by P in t hours is equal to P's speed multiplied by t. Similarly, the distance covered by Q in (t - p) hours is equal to Q's speed multiplied by (t - p), and the distance covered by R in (t + q) hours is equal to R's speed multiplied by (t + q).
Now, let's calculate the distances covered by P, Q, and R in terms of t:
Distance covered by P = P's speed multiplied by t = a * t
Distance covered by Q = Q's speed multiplied by (t - p) = (a + b) * (t - p)
Distance covered by R = R's speed multiplied by (t + q) = (a + 2b) * (t + q)
Since we want Q and R to overtake P at the same time, their distances covered should be equal. Therefore, we have the following equation:
(a + b) * (t - p) = (a + 2b) * (t + q)
Now, let's solve this equation to find the value of q in terms of p:
(a + b) * (t - p) = (a + 2b) * (t + q)
(a + b) * t - (a + b) * p = (a + 2b) * t + (a + 2b) * q
By simplifying the equation, we get:
(a + b) * t - (a + 2b) * t = (a + 2b) * q + (a + b) * p
a * t + b * t - (a * t + 2b * t) = (a + 2b) * q + (a + b) * p
-a * t - b * t = (a + 2b) * q + (a + b) * p
(-a - b) * t = (a + 2b) * q + (a + b) * p
-t = (a + 2b) * q + (a + b) * p
Finally, to find the value of q in terms of p, we divide both sides of the equation by (a + 2b):
q = (-t - (a + b) * p) / (a + 2b)
Therefore, q is equal to (-t - (a + b) * p) divided by (a + 2b).
Velocity = distance/time
P=a = x/t1
Q=a+b = x/t2
R=a+2b = x/t3
Q stats p hours after P:
(a+b)=X/(t1+p)
hours after Q should R start:
a+2b = X/(t2+h)
t1=x/a
t2=t1+p
t3=t1+p+h
Velocity Q = R
by manipulating the velocity formula:
Velocity/time = distance
Q distance = R distance
Velocity*time=distance:
(a+b)/(t1+p) = (a+2b)/(t1+p+h)
solve for h: