A truck is stopped at a stoplight. When the light turns green, it accelerates at 2.5 m/s². At the same instant, a car passes the truck going 15 m/s. Where and when does the truck catch up with the car?
First I found out how long it took the truck to reach 15 m/s. Then I found out how far the truck had traveled in the time it took to reach 15 m/s. Then I found out how far the car had traveled in the time it took the truck to reach 15 m/s.
A version of the question was posted here and was answered incorrectly.
The cars do not pass when the accelerating car reaches 15 m/s. The other car is still far ahead at that time.
The cars pass when
15 t = (1/2)at^2 = 1.25 t^2
t = 15/1.25 = 12 s
Distance = 15 t = 180 m
To find out where and when the truck catches up with the car, let's break the problem down into steps:
Step 1: Find the time it takes for the truck to reach the car's initial speed.
To do this, we can use the equation of motion: v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
Given:
Initial speed of the car (u) = 15 m/s
Acceleration of the truck (a) = 2.5 m/s²
Since the truck starts from rest, its initial velocity (u) is 0.
So, the equation becomes:
15 m/s = 0 + (2.5 m/s²) * t
Solving for t:
15 m/s = 2.5 m/s² * t
t = 15 m/s / 2.5 m/s²
t = 6 seconds
Step 2: Find the distance traveled by the truck in 6 seconds.
We can use the equation of motion again to find the distance (s) covered by the truck:
s = ut + (1/2) * a * t²
Given:
Acceleration of the truck (a) = 2.5 m/s²
Time (t) = 6 seconds
Using the equation:
s = (0 m/s) * (6 s) + (1/2) * (2.5 m/s²) * (6 s)²
s = 0 + (1/2) * 2.5 m/s² * 36 s²
s = (1/2) * 2.5 m/s² * 36 s²
s = (1/2) * 90 m
s = 45 meters
Step 3: Find the distance traveled by the car in 6 seconds.
Since the car maintains a constant speed of 15 m/s throughout, we can simply use the formula: distance = speed * time.
Given:
Speed of the car = 15 m/s
Time (t) = 6 seconds
Therefore, the distance traveled by the car is:
s = 15 m/s * 6 s
s = 90 meters
Step 4: Determine the point of intersection.
To find where the truck catches up with the car, we need to compare the distances covered by both.
Since the truck traveled 45 meters and the car traveled 90 meters, the truck catches up with the car at a point when it has covered the same distance.
So, the truck catches up with the car at a distance of 45 meters from the starting point.
To determine the time when this happens, we can use either the time it took for the truck to reach 15 m/s (6 seconds) or the time it took for the car to travel 45 meters (which should be the same).
Therefore, the truck catches up with the car after 6 seconds, exactly 45 meters from the starting point.
To solve this problem, we will need to calculate the time it takes for the truck to reach the same speed as the car and then use that time to determine where the two vehicles meet.
1. Calculate the time it takes for the truck to reach 15 m/s:
To find the time it takes for the truck to reach the same speed as the car, we can use the formula: time = (final velocity - initial velocity) / acceleration.
Given that the initial velocity of the truck is 0 m/s and the acceleration is 2.5 m/s², we can substitute these values into the formula:
time = (15 m/s - 0 m/s) / 2.5 m/s² = 6 seconds.
2. Calculate the distance traveled by the truck in that time:
To find the distance traveled by the truck, we can use the formula: distance = initial velocity * time + 0.5 * acceleration * time^2.
The initial velocity of the truck is 0 m/s, the time is 6 seconds, and the acceleration is 2.5 m/s². Substituting these values into the formula:
distance = 0 m/s * 6 s + 0.5 * 2.5 m/s² * (6 s)^2 = 45 meters.
3. Calculate the distance traveled by the car in the same time:
Since the car has already been traveling at a constant speed of 15 m/s, we can calculate the distance traveled using the formula: distance = velocity * time.
The velocity of the car is 15 m/s and the time is 6 seconds, so:
distance = 15 m/s * 6 s = 90 meters.
Therefore, the truck catches up with the car after traveling a distance of 45 meters, while the car has traveled a distance of 90 meters.
To determine where and when the truck catches up with the car, we need to consider their respective locations before they meet. Since the car is already ahead of the truck, it will take additional time for the truck to catch up.
Let's say the initial distance between the truck and the car is d1 meters. After the calculated time of 6 seconds, the truck will have traveled 45 meters and the car will have traveled 90 meters. At this point, the relative distance between them has decreased by 45 meters (90 meters - 45 meters).
Therefore, the truck catches up with the car when the distance between them (d2) becomes equal to or less than d1 - 45 meters. At this point, the truck and car will both be 45 meters away from their initial positions.
To find the exact location and time, we need more information about the initial distance between them or any other relevant details.