A red car and a green car, identical except for color, move toward each other in adjacent lanes and parallel to an x-axis. At time t=0, the red car is at x sub r = 0 and the green car is at x sub g = 220m. If the red car has a constant velocity of 20km/h, the cars pass each other at x = 44.5m, and if it has a constant velocity of 40 km/h, they pass each other at x=76.6m. What are (a) the initial velocity and (b) the constant acceleration of the green car?
To solve this problem, we can use the concepts of relative motion. Let's break down the problem step by step:
Step 1: Convert the velocities (given in km/h) to m/s to maintain consistent units.
The velocity of the red car is given as 20 km/h. To convert this to m/s, we can use the conversion factor: 1 km/h = 0.277778 m/s.
So, the velocity of the red car is:
v_r = 20 km/h * 0.277778 m/s = 5.55556 m/s
Step 2: Calculate the time it takes for the cars to pass each other at each given distance.
Using the formula: distance = velocity * time, we can rearrange it to solve for time:
time = distance / velocity
For the first scenario, when the red car has a velocity of 5.55556 m/s and they pass each other at x = 44.5 m:
time_1 = 44.5 m / 5.55556 m/s ≈ 8 s
For the second scenario, when the red car has a velocity of 40 km/h and they pass each other at x = 76.6 m:
time_2 = 76.6 m / (40 km/h * 0.277778 m/s) ≈ 3.46 s
Step 3: Use the time values to find the initial velocity of the green car.
We know the initial position of the green car (x_g = 220 m) and the time it takes for the cars to pass each other at each distance. We can use the formula: velocity = distance / time
For the first scenario:
v_g_1 = (44.5 m - 0 m) / 8 s = 5.5625 m/s
For the second scenario:
v_g_2 = (76.6 m - 0 m) / 3.46 s ≈ 22.1316 m/s
Therefore, the initial velocities of the green car for the two scenarios are:
(a) For the red car velocity of 20 km/h: v_g_1 = 5.5625 m/s
(b) For the red car velocity of 40 km/h: v_g_2 ≈ 22.1316 m/s
Step 4: Calculate the constant acceleration of the green car.
To find the constant acceleration, we can use the formula: acceleration = (final velocity - initial velocity) / time.
For the first scenario:
a_1 = (0 m/s - 5.5625 m/s) / 8 s = -0.6953 m/s^2
For the second scenario:
a_2 = (0 m/s - 22.1316 m/s) / 3.46 s ≈ -6.38 m/s^2
Therefore, the constant accelerations of the green car for the two scenarios are:
(a) For the red car velocity of 20 km/h: a_1 ≈ -0.6953 m/s^2
(b) For the red car velocity of 40 km/h: a_2 ≈ -6.38 m/s^2
In summary:
(a) The initial velocity of the green car is approximately 5.5625 m/s if the red car velocity is 20 km/h.
(b) The initial velocity of the green car is approximately 22.1316 m/s if the red car velocity is 40 km/h.
(a) The constant acceleration of the green car is approximately -0.6953 m/s^2 if the red car velocity is 20 km/h.
(b) The constant acceleration of the green car is approximately -6.38 m/s^2 if the red car velocity is 40 km/h.
To solve this problem, we need to use the equations of motion along with the given information. Let's break down the problem step by step.
Step 1: Determine the time it takes for the cars to pass each other.
We know that the distance traveled by both cars is the same when they pass each other. Therefore, we can set up the equation:
Distance traveled by the red car = Distance remaining for the green car
Since the red car starts from x=0 and the green car starts from x=220m, the distance traveled by the red car is x and the distance remaining for the green car is (220 - x). Substituting the given values:
x = 220 - x
2x = 220
x = 110 m
We know that distance = velocity × time (d = vt). Let's use this equation to find the time it takes for the cars to pass each other when the red car has a velocity of 20 km/h.
110 m = (20 km/h) × (time)
Converting 20 km/h to m/s:
20 km/h = (20 × 1000 m) / (3600 s) = 5.56 m/s
Now we can determine the time it takes:
110 m = 5.56 m/s × time
time = 110 m / 5.56 m/s
time = 19.78 s
Similarly, when the red car has a velocity of 40 km/h, the time it takes for the cars to pass each other is:
110 m = (40 km/h) × (time)
Converting 40 km/h to m/s:
40 km/h = (40 × 1000 m) / (3600 s) = 11.11 m/s
time = 110 m / 11.11 m/s
time = 9.91 s
Step 2: Calculate the initial velocity of the green car.
The initial velocity of the green car is the velocity at t=0, when it starts from x=220m.
Using the equation of motion: x = ut + (1/2)at^2
At t=0, x=220m, and since the green car starts from rest, the initial velocity (u) is 0.
220 m = (0 m/s) × t + (1/2) a (t^2)
220 m = (1/2) a (t^2)
t^2 = (440 m) / a
When the red car has a velocity of 20 km/h, the time is t = 19.78 s (as calculated above). Substituting these values:
(19.78 s)^2 = (440 m) / a
391.21 s^2 = 440 m / a
a = (440 m) / (391.21 s^2)
a ≈ 1.123 m/s^2
Similarly, when the red car has a velocity of 40 km/h, the time is t = 9.91 s. Substituting these values:
(9.91 s)^2 = (440 m) / a
98.03 s^2 = 440 m / a
a = (440 m) / (98.03 s^2)
a ≈ 4.488 m/s^2
Therefore, the answers to the questions are:
(a) The initial velocity of the green car is 0 m/s.
(b) The constant acceleration of the green car is approximately 1.123 m/s^2 when the red car's velocity is 20 km/h, and approximately 4.488 m/s^2 when the red car's velocity is 40 km/h.