Car a starts from rest and speeds up with constant accelleration until it reaches its max speed. Car b starts from rest, at the moment car a is at max speed, and speeds up with constant acceleration. How much time elapses between when car b starts moving, and car b passes car a.
I have tried to solve using kinematic equations but nothing seems to work.
Car A
vA
aA
tA
Car B
vB
aB
tB
To solve this problem, we can use the equations of motion, also known as the kinematic equations. Let's break down the problem and solve it step by step.
Given:
Car A:
Initial velocity (uA) = 0 (starts from rest)
Final velocity (vA) = max speed
Acceleration (aA) = constant
Car B:
Initial velocity (uB) = 0 (starts from rest)
Final velocity (vB) = ?
Acceleration (aB) = constant
We need to find the time it takes for Car B to reach and pass the max speed of Car A.
Step 1: Determine the time it takes for Car A to reach the max speed.
We can use the first kinematic equation:
vA = uA + aA*tA
Since Car A starts from rest (uA = 0), the equation simplifies to:
vA = aA*tA
Solving for tA:
tA = vA / aA
Step 2: Calculate the additional time it takes for Car B to reach the same maximum speed.
Since Car B starts at the same initial position where Car A has already reached max speed, Car B needs to catch up to Car A's speed without considering the distance covered by Car A during this time.
The relative speed of Car B with respect to Car A would be:
Relative speed (vB_relative) = vA
Using the second kinematic equation:
vB = uB + aB*tB
Since Car B starts from rest (uB = 0), the equation simplifies to:
vB = aB*tB
We know that at the moment Car B starts moving, Car A is already at max speed, so the relative speed of Car B with respect to Car A is vA.
vB_relative = vB - vA = 0 (since vB = vA)
Step 3: Determine the time it takes for Car B to reach the same maximum speed:
0 = aB*tB
Since the acceleration is non-zero (aB ≠ 0), the only solution is tB = 0.
So, when Car B starts moving, it immediately matches Car A's maximum speed.
Therefore, the answer is that car B passes car A at the moment it starts moving. No additional time elapses between when Car B starts moving and when it passes Car A.
Da = Db same distance
total time a = time b + delay b
we want time b
Car A
accelerate phase:
vi = aA delayb
Di = .5 aA delayb^2
constant speed phase
Da = Di + vi t
= .5 aA delayb^2 + aA delayb timeb
Car B
Db = .5 aB timeb^2
now Da=Db so
.5 aA delayb^2 + aA delayb timeb = .5 aB timeb^2
Now you really need to know what that max speed vi is to get the time car A accelerates which is delay B
or just put it in here as
delayb = vi/aA
.5 vi^2/aA + vi timeb = .5 aB timeb^2
solver quadratic for timeb