In a relay race, runner A is carrying a baton and has a speed of 3.0 m/s . When he is 25M behind the starting line, runner B starts from rest and accelerates at .10 m/s^2. How long afterwards will A catch up with B to pass the baton to B?
distance covered by behind runner: 3*t
distance by starter: 1/2 a t^2
these times have to be equal,
so 3t-25=1/2(.1)t^2
multiply by 20, put in quadratic form
t^2-60t-500=0
t= (30-+ sqrt(3600/4 +500))
= 30 +- 10 sqrt (9+5)
now for an engneering check:
let sqrt (15) appx=4
t=70, or -10 throw out the negative time, we are not going backwards.
Distance covered by runner: 280
distance covered by starter: 4900*.05=appx 250 which sounds like a reasonable answer, as runner has to go 25 m further.
Work it accurately.
To solve this problem, we can use the equation of motion:
d = v0*t + (1/2)*a*t^2
where:
d is the distance traveled,
v0 is the initial velocity,
a is the acceleration, and
t is the time.
First, let's find the time it takes for Runner B to reach the starting line.
Using the equation of motion for Runner B:
25 = 0*t + (1/2)*0.10*t^2
Dividing by 0.10 on both sides, we get:
25/0.10 = t^2
Simplifying the equation, we have:
250 = t^2
Taking the square root of both sides, we get:
t ≈ 15.81 seconds
Now, let's find the distance traveled by Runner A in this time.
Using the equation for displacement:
d = v0*t
d = 3.0 m/s * 15.81 s
d ≈ 47.43 meters
Since Runner A starts 25 meters behind, the total distance covered by Runner A is: 47.43 + 25 = 72.43 meters.
Now, let's find the time it takes for Runner A to catch up to Runner B.
Using the equation of motion for Runner A:
d = v0*t
72.43 = 3.0*t
Dividing by 3.0 on both sides, we get:
t ≈ 24.14 seconds
Therefore, it will take approximately 24.14 seconds for Runner A to catch up with Runner B and pass the baton.
To solve this problem, we need to calculate the time it takes for runner A to catch up with runner B. Let's break down the problem into steps:
Step 1: Determine the initial distance between runner A and B:
The initial distance between runner A and B is given as 25 meters.
Step 2: Calculate the time it takes for runner A to reach the same position as runner B:
To calculate the time taken by runner A to catch up with runner B, we need to consider their relative velocities. The relative velocity is the difference between their velocities.
The velocity of runner A is given as 3.0 m/s, while runner B starts from rest and accelerates at 0.10 m/s². To compare their velocities, we need to find the velocity of runner B when runner A catches up.
We can calculate the velocity of runner B using the formula:
v = u + at
where:
v = final velocity
u = initial velocity (in this case, 0 m/s)
a = acceleration (0.10 m/s² from the problem)
t = time taken
Since runner B starts from rest (u = 0 m/s), the formula simplifies to:
v = at
Plugging in the values, we get:
v = 0.10 m/s² × t
Step 3: Equate the distances traveled by runner A and runner B:
At the point of catching up, the distance traveled by runner A is the same as the distance traveled by runner B.
The distance traveled by runner A can be calculated using the formula:
distance = speed × time
where:
distance = distance traveled by runner A
speed = velocity of runner A (3.0 m/s)
time = time taken by runner A to catch up
The distance traveled by runner B can be calculated using the formula:
distance = 0.5 × acceleration × time²
where:
distance = distance traveled by runner B
acceleration = acceleration of runner B (0.10 m/s²)
time = time taken by runner B to be caught up by runner A
Step 4: Set up the equation:
Equating the distances traveled by runner A and runner B, we have:
distance traveled by A = distance traveled by B
(speed of A) × (time taken by A) = (0.5 × acceleration of B × time taken by B²)
Substituting the given values:
(3.0 m/s) × (time taken by A) = (0.5 × 0.10 m/s² × time taken by B²)
Now we need to solve this equation to find the time taken by A to catch up with B.
Step 5: Solve the equation:
3.0 × time taken by A = 0.05 × time taken by B²
Dividing both sides by 0.05:
60 × time taken by A = time taken by B²
In this equation, we need to find the matching time taken by runner A and runner B. Let's assume both runners are denoted by the variable "t". So the equation can be rewritten as:
60t = t²
Step 6: Solve the quadratic equation:
To solve the quadratic equation, let's rearrange it:
t² - 60t = 0
Factorizing it, we get:
t(t - 60) = 0
Therefore, either t = 0 (which is not possible in this context) or t - 60 = 0.
Solving for t:
t - 60 = 0
t = 60
So the time taken by runner A to catch up with runner B is 60 seconds.
Therefore, in 60 seconds, runner A will catch up with runner B to pass the baton.