A brown road runner is standing still pecking at soil looking for worms. Suddenly, a black road runner travelling at a constant speed of 9.25 m/s, overtakes the brown roadrunner. The brown road runner notices that the other one has a worm in its mouth and begins to accelerate at 2.75 m/s2 to catch up to and steal the worm. How far will the brown road runner have to travel to catch up to the black road runner? Assume the stationary brown road runner begins to move the moment the black road runner passes it.
The would have to have travelled the same distance in the same lapse of time.
Let time=t, then
Brown roadrunner would have travelled
S1=ut+(1/2)at^2
=0(t)+(1/2)2.75t^2
=1.375t^2
Black roadrunner would have travelled
S2=9.25t
Both distances S1 and S2 are in metres.
For them to meet, S1=S2, =>
1.375t^2=9.25t =>
t(9.25-1.375t)=0
Solve for t.
t=0 (when black roadrunner past brown), or
(9.25-1.375t)=0
=> t=6.727 s.
S1=S2=9.25(6.727)=62.23 m
9.25 t = 1/2 * 2.75 * t^2
1.375 t^2 - 9.25 t = 0
t (1.375 t - 9.25) = 0
t = 0 ... when the black one passes
t = 9.25 / 1.375 ... brown catches up
To find the distance the brown road runner has to travel to catch up to the black road runner, we need to consider the relative speeds and acceleration of the two road runners.
Let's break down the problem step by step:
Step 1: Determine the initial distance between the road runners.
Since the black road runner overtakes the brown road runner while traveling at a constant speed, the initial distance between them is the same as the distance the black road runner travels before overtaking the brown road runner.
Step 2: Calculate the time it takes for the brown road runner to catch up.
To calculate this, we need to find the time it takes for the brown road runner to accelerate and reach the same speed as the black road runner. We can use the formula of motion:
v = u + at
Here, v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
For the brown road runner:
Final velocity (v) = 9.25 m/s (same as the black road runner)
Initial velocity (u) = 0 m/s (since it starts from rest)
Acceleration (a) = 2.75 m/s^2
Plugging these values into the equation, we can calculate the time needed for the brown road runner to catch up.
Step 3: Calculate the distance the brown road runner travels during acceleration.
To find this distance, we can use the equation of motion:
s = ut + (1/2)at^2
where s is the distance, u is the initial velocity, a is the acceleration, and t is the time.
For the brown road runner:
Initial velocity (u) = 0 m/s
Acceleration (a) = 2.75 m/s^2
Time (t) is the time calculated in the previous step.
Substituting these values into the formula, we can calculate the distance traveled by the brown road runner during acceleration.
Step 4: Calculate the total distance traveled by the brown road runner.
To find the total distance, we add the distance traveled during acceleration to the initial distance between the two road runners.
Now, let's calculate the distance the brown road runner needs to travel:
Step 1: Calculate the initial distance.
The initial distance is equal to the distance traveled by the black road runner before overtaking the brown road runner. Since we are not given this distance, we cannot calculate it.
Unfortunately, without the initial distance, we are unable to proceed with the calculation. Please provide the distance traveled by the black road runner before overtaking the brown road runner, and we will be able to solve the problem.