At the instant the traffic light turns green, an automobile that has been waiting at an intersection starts ahead with a constant acceleration of 2.50 . At the same instant, a truck, traveling with a constant speed of 15.4 , overtakes and passes the automobile.
How far beyond its starting point does the automobile overtake the truck?
How fast is the automobile traveling when it overtakes the truck?
Write equations for horizontal distance travelled vs time for auto and truck. Then set equal to each other and solve for the time when passing occurs. Knowing that time, you can also solve for the velocity of the accelerating auto.
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To solve this problem, we can start by analyzing the motion of both the automobile and the truck.
Let's define the following variables:
- t: time taken for the automobile to overtake the truck (in seconds)
- a: acceleration of the automobile (2.50 m/s^2)
- v0: initial velocity of the automobile (0 m/s)
- vt: velocity of the truck (15.4 m/s)
- d: distance traveled by the automobile when it overtakes the truck (in meters)
- at: distance traveled by the truck when it is overtaken by the automobile (in meters)
For the automobile, we can use the equation of motion:
d = v0t + (1/2)at^2
Since the initial velocity of the automobile is 0, the equation simplifies to:
d = (1/2)at^2
For the truck, the distance traveled is simply:
at = vt * t
Since the automobile overtakes the truck, we know that at = d.
Setting these two equations equal to each other, we have:
(1/2)at^2 = vt * t
Simplifying this equation, we get:
(1/2)a * t^2 = vt * t
Now, let's solve for t:
(1/2) * 2.50 * t^2 = 15.4 * t
Simplifying further:
1.25 * t^2 = 15.4 * t
Dividing both sides by t:
1.25 * t = 15.4
Solving for t:
t = 15.4 / 1.25
t ā 12.32 seconds
Now that we know the time it takes for the automobile to overtake the truck, we can use this time to find the distance traveled by the automobile:
d = (1/2) * a * t^2
d = (1/2) * 2.50 * (12.32)^2
d ā 190.75 meters
Therefore, the automobile overtakes the truck approximately 190.75 meters beyond its starting point.
To find the velocity of the automobile when it overtakes the truck, we can use the equation of motion:
v = v0 + at
Substituting the values:
v = 0 + 2.50 * 12.32
v ā 30.8 m/s
Therefore, the automobile is traveling at approximately 30.8 m/s when it overtakes the truck.
Remember, it is important to understand the concepts and solve the problem step by step, rather than just relying on the final answer.