Two cars are traveling along a straight road. Car A maintains a constant speed of 79 km/h and car B maintains a constant speed of 114 km/h. At t = 0, car B is 36 km behind car A.
(a) How much farther will car A travel before car B overtakes it?
? km
(b) How much ahead of A will B be 30 s after it overtakes A?
Please use kinematics equations! Thank you!!
(a) How much farther will car A travel before car B overtakes it?
Let's assume that car B overtakes car A after t hours.
The distance traveled by car A in t hours is given by:
Distance_A = Speed_A * Time
Distance_A = 79 km/h * t
The distance traveled by car B in t hours is given by:
Distance_B = Speed_B * Time
Distance_B = 114 km/h * t
At the moment of overtaking, the distance traveled by car B will be the same as the distance traveled by car A plus the initial separation distance between them:
Distance_B = Distance_A + Separation Distance
114 km/h * t = 79 km/h * t + 36 km
Now we can solve for t:
35 km/h * t = 36 km
t = 1.03 hours
To find how much farther car A will travel, we substitute the value of t into the equation for Distance_A:
Distance_A = 79 km/h * 1.03 hours
Distance_A = 81.37 km
Therefore, car A will travel approximately 81.37 km farther before car B overtakes it.
(b) How much ahead of A will B be 30 s after it overtakes A?
Since car B overtakes car A at time t = 1.03 hours, we first need to convert 30 seconds into hours:
30 seconds = 30/3,600 hours
t' = 0.0083 hours
After t' hours, the distance traveled by car A is given by:
Distance_A' = Speed_A * Time
Distance_A' = 79 km/h * 0.0083 hours
After t' hours, the distance traveled by car B is given by:
Distance_B' = Speed_B * Time
Distance_B' = 114 km/h * 0.0083 hours
To find how much ahead of car A car B will be, we subtract the distance traveled by car A from the distance traveled by car B:
Distance_B' - Distance_A' = (114 km/h * 0.0083 hours) - (79 km/h * 0.0083 hours)
Therefore, car B will be approximately ____ km ahead of car A 30 seconds after overtaking it. (Oops, my humor circuits need some adjustment.)
To solve this problem, we can use the kinematic equations of motion. Let's find the time it takes for car B to catch up to car A first.
Let's assume:
Initial position of car A = 0 km
Initial position of car B = -36 km (since car B is 36 km behind car A)
Relative velocity of car B with respect to car A = 114 km/h - 79 km/h = 35 km/h
To convert the speeds to the same unit, we can use the conversion factor: 1 km/h = 1/3.6 m/s.
Relative velocity of car B with respect to car A = 35 km/h × (1/3.6 m/s)/(1 km/h) = 35 × (1/3.6) m/s ≈ 9.72 m/s
Using the equation for distance traveled, s = ut + (1/2)at^2, where u is the initial velocity, t is the time, and a is the acceleration (which is 0 since the cars are maintaining constant speeds), we can solve for t.
Using the equation s = ut, we can rewrite it as:
36 km = (9.72 m/s) × t
Converting the distance to meters: 36 km × 1000 m/km = 36000 m
36000 m = 9.72 m/s × t
Solving for t, we get:
t = 36000 m / 9.72 m/s ≈ 3703.3 seconds
(a) How much farther will car A travel before car B overtakes it?
To find the distance car A will travel before being overtaken, we can use the equation s = ut, where u is the constant velocity of car A.
s = (79 km/h) × t
Converting the velocity to meters per second:
s = (79 km/h) × (1/3.6 m/s)/(1 km/h) × t
Substituting the value of t we found earlier:
s = (79 km/h) × (1/3.6 m/s)/(1 km/h) × 3703.3 s
Simplifying:
s ≈ 79196.7 m
Therefore, car A will travel approximately 79196.7 meters farther before car B overtakes it.
(b) How much ahead of A will B be 30 seconds after it overtakes A?
To find how far ahead car B will be after overtaking car A, we can use the equation s = ut, where u is the constant velocity of car B.
s = (114 km/h) × t
Converting the velocity to meters per second:
s = (114 km/h) × (1/3.6 m/s)/(1 km/h) × t
Substituting the value of t we found earlier:
s = (114 km/h) × (1/3.6 m/s)/(1 km/h) × 3703.3 s
Simplifying:
s ≈ 113612.9 m
Therefore, car B will be approximately 113612.9 meters ahead of car A after 3703.3 seconds, which is the same as 30 seconds.
To solve this problem, we can use the kinematics equations of motion. Let's break down the problem into two parts: first, calculating the time it takes for car B to overtake car A, and secondly, calculating the distance and position after the overtake.
Step 1: Calculating the time of overtake
In this step, we need to find the time at which Car B overtakes Car A. We know that Car A maintains a constant speed of 79 km/h, and Car B maintains a constant speed of 114 km/h. At t = 0, Car B is 36 km behind Car A.
Let's set up equations to represent the distance traveled by each car:
Distance traveled by Car A = 79t
Distance traveled by Car B = 114t + 36
To find the time of overtake, we need to set these two equations equal to each other:
79t = 114t + 36
Now, let's solve for t:
79t - 114t = 36
-35t = 36
t = -36/35
t ≈ -1.0286 hours
Since time cannot be negative, we'll take the positive solution:
t ≈ 1.0286 hours
Step 2: Calculating the distance and position after overtake
Now that we know the time it takes for Car B to overtake Car A, we can calculate the distance traveled by Car A and Car B at that time.
Distance traveled by Car A = 79 km/h * 1.0286 h
Distance traveled by Car B = 114 km/h * 1.0286 h + 36 km
Calculate the above equations to get the respective distance values.
(a) How much farther will Car A travel before Car B overtakes it?
To find the difference in distance traveled by both cars, subtract the distance traveled by Car A from the distance traveled by Car B at the time of overtake.
Difference in distance = Distance traveled by Car B - Distance traveled by Car A
(b) How much ahead of A will B be 30 s after it overtakes A?
To find the position of Car B relative to Car A, we need to find the position of Car A after 30 seconds and subtract it from the total distance traveled by Car B at the time of overtake.
Position of Car A after 30 seconds = 79 km/h * (30/3600) h
Position of Car B after overtake = Distance traveled by Car B at the time of overtake - Position of Car A after 30 seconds
Calculate the above equation to get the position of Car B relative to Car A after 30 seconds.
By applying the above steps, you can find the solutions to the given questions.
25
the only equation you really need is d=vt, or t=d/v. That gives you
114t = 79t+36
35t=36
t = 36/35
That's how long it takes B to overtake A.
In that time, A travels 79*36/35 = 81.26 km
You can think of this another way by noting that B is traveling 35 km/hr faster than A. So, it just takes 36/35 hours to make up the 35 km.
(b) Knowing that B travels 35 km/hr faster than A, it's easy to figure how much ahead it is after 30s = 1/120 hr