In the figure here, a red car and a green car move toward each other in adjacent lanes and parallel to an x axis. At time t = 0, the red car is at xr = 0 and the green car is at xg = 223 m. If the red car has a constant velocity of 22.0 km/h, the cars pass each other at x = 44.7 m. On the other hand, if the red car has a constant velocity of 44.0 km/h, they pass each other at x = 76.6 m. What are (a) the initial velocity (in km/h) and (b) the (constant) acceleration (in m/s2) of the green car
a=2.1m/s^2 vo=13.55m/s
To find the initial velocity (a) and the constant acceleration (b) of the green car:
Let's use the equation of motion for constant acceleration:
x = xo + vo⋅t + 0.5⋅a⋅t^2
Where:
x is the position of the car at a given time,
xo is the initial position of the car,
vo is the initial velocity of the car, and
a is the constant acceleration of the car.
We know that the cars pass each other at two different positions: x = 44.7 m and x = 76.6 m.
For x = 44.7 m:
At this position, the red car has a velocity of 22.0 km/h. Since the green car is moving towards the red car, its velocity is negative. So, we can write:
xr + xg = 44.7 m
For the red car:
xo = 0
vo = 22.0 km/h
t = unknown
a = unknown
For the green car:
xo = 223 m
vo = -unknown (negative because it's moving towards the red car)
t = unknown
a = unknown
Converting the velocity of the red car to m/s:
vo_red = 22.0 km/h = 22.0 * (1000/3600) m/s = 6.11 m/s
From the equation of motion, we can write:
0 + (6.11 m/s)⋅t + 0.5⋅a_green⋅t^2 = 44.7 m
(1)
For x = 76.6 m:
At this position, the red car has a velocity of 44.0 km/h. So, we can write:
xr + xg = 76.6 m
For the red car:
xo = 0
vo = 44.0 km/h
t = unknown
a = unknown
For the green car:
xo = 223 m
vo = -unknown (negative because it's moving towards the red car)
t = unknown
a = unknown
Converting the velocity of the red car to m/s:
vo_red = 44.0 km/h = 44.0 * (1000/3600) m/s = 12.22 m/s
From the equation of motion, we can write:
0 + (12.22 m/s)⋅t + 0.5⋅a_green⋅t^2 = 76.6 m
(2)
Now, we have a system of two equations with two unknowns (t and a_green). We can solve these equations simultaneously to find the initial velocity (a) and the constant acceleration (b) of the green car.
To determine the initial velocity and constant acceleration of the green car, we can use the two pieces of information given about the positions of the cars at different times and the constant velocity of the red car.
Let's start by using the information given with the red car's constant velocity of 22.0 km/h and the point of passing at x = 44.7 m. We can derive the following equations:
1. red car: xr = ut (where xr is the position of the red car, u is its initial velocity, and t is time)
2. green car: xg = 223 - ut (where xg is the position of the green car, considering it's moving towards the red car)
3. xg + xr = 44.7 (this equation represents the point of passing)
Using equation 1, we have:
0 + 22t = 44.7 (since the red car starts at xr = 0 and its initial velocity is given as 22.0 km/h)
Simplifying this equation, we find:
t = 44.7 / 22 ≈ 2.04 seconds
We now have the time it takes for the cars to pass each other when the red car's velocity is 22.0 km/h.
Next, we can find the initial velocity (u) and constant acceleration (a) of the green car using the equation 2.
From equation 2, substituting the values we know, we have:
xg = 223 - u * t
Solving for u (initial velocity of the green car):
u = (223 - xg) / t
Substituting the values we know:
u = (223 - 44.7) / 2.04 ≈ 88.20 km/h
So, the initial velocity of the green car when the red car's velocity is 22.0 km/h is approximately 88.20 km/h.
Now, to calculate the acceleration of the green car, we can use equation 2 again:
xg = 223 - u * t
Differentiating both sides of the equation with respect to time (t), we get:
dg/dt = -u
This means the velocity v of the green car is constant as long as there is no acceleration.
Therefore, the acceleration of the green car in this scenario is 0 m/s^2.
In summary:
(a) The initial velocity of the green car when the red car's velocity is 22.0 km/h is approximately 88.20 km/h.
(b) The acceleration of the green car is 0 m/s^2.