The car stationary in front of a red traffic light. As the light turns green , a truck goes past at a constant velocity of 1.5 m/s. At the same moment , the car begins to accelerate at 1.25 m/s 2 . When it reaches 2.5 m/s the car continues at this velocity . When does the car pass the truck ? How far they gone from the traffic light at that time ? 25 s / 0.38 km
(1/2)(1.25)(2^2) + 2.5(t-2) = 1.5t
t = 2.5
2.5*1.5 = 3.75
If the 25s/0.38km is supposed to be the answer, I suspect the units have gotten garbled.
To determine when the car passes the truck and how far they have gone from the traffic light at that time, we need to calculate the time it takes for the car to reach the same velocity as the truck.
Let's break down the problem step by step:
1. First, let's find the time it takes for the car to reach a velocity of 2.5 m/s. We can use the formula 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 velocity (u) of the car = 0 m/s (since it was stationary)
- Acceleration (a) of the car = 1.25 m/s^2
- Final velocity (v) of the car = 2.5 m/s
Plugging in these values into the formula, we get:
2.5 = 0 + 1.25t
Solving for t:
t = 2.5 / 1.25
= 2 seconds
So it takes the car 2 seconds to reach a velocity of 2.5 m/s.
2. Next, let's find out how far the car and truck have traveled during this time.
Since the truck is moving at a constant velocity of 1.5 m/s, we can use the formula of motion:
s = ut + (1/2)at^2
For the truck:
- Initial velocity (u) of the truck = 1.5 m/s
- Acceleration (a) of the truck = 0 m/s^2 (because it's moving at a constant velocity, no acceleration)
- Time (t) = 2 seconds (the time it took for the car to reach a velocity of 2.5 m/s)
Plugging in these values, we get:
s_truck = (1.5 * 2) + (0 * (2^2))
= 3 meters
The truck has traveled 3 meters during this time.
For the car:
- Initial velocity (u) of the car = 0 m/s (since it was stationary)
- Acceleration (a) of the car = 1.25 m/s^2
- Time (t) = 2 seconds (the same as the truck)
Plugging in these values, we get:
s_car = (0 * 2) + (0.5 * 1.25 * (2^2))
= 2.5 meters
The car has traveled 2.5 meters during this time.
3. Lastly, let's calculate how far they have gone from the traffic light at that time.
The distances traveled by the car and the truck are relative to the traffic light, so we can simply add them up:
Total distance from the traffic light = s_truck + s_car
= 3 + 2.5
= 5.5 meters
Note that to convert this distance to kilometers, you can divide it by 1000:
Total distance from the traffic light = 5.5 meters / 1000
= 0.0055 kilometers
≈ 0.006 km (rounded to three decimal places)
Therefore, when the car passes the truck, they have gone approximately 0.006 kilometers (or 6 meters) from the traffic light. This happens 2 seconds after the car starts accelerating.
To find out when the car passes the truck and how far they have gone from the traffic light at that time, we need to understand the motion of both vehicles. Let's break it down step-by-step:
Step 1: Determine the time it takes for the car to reach the truck's velocity.
Using the formula for acceleration:
v = u + at
where:
v = final velocity (2.5 m/s)
u = initial velocity (0 m/s)
a = acceleration (1.25 m/s^2)
t = time
To find t, we rearrange the formula:
t = (v - u) / a
Substituting the values:
t = (2.5 - 0) / 1.25
t = 2 seconds
So it takes the car 2 seconds to reach a velocity of 2.5 m/s.
Step 2: Calculate the distance traveled by the car and the truck in those 2 seconds.
Using the formula for distance with constant acceleration:
s = ut + (1/2)at^2
where:
s = distance
u = initial velocity
t = time
a = acceleration
For the car:
u = 0 m/s (initial velocity)
t = 2 seconds (time)
a = 1.25 m/s^2 (acceleration from the car)
Substituting the values:
s_car = (0 * 2) + (1/2) * 1.25 * (2^2)
s_car = 0 + 0.625 * 4
s_car = 2.5 meters
For the truck:
u = 1.5 m/s (constant velocity)
t = 2 seconds (time)
Substituting the values:
s_truck = 1.5 * 2
s_truck = 3 meters
Therefore, in the first 2 seconds, the car travels 2.5 meters while the truck travels 3 meters.
Step 3: Calculate the time it takes for the car to catch up to the truck.
Since the truck maintains a constant velocity, we can determine how long it takes for the car to bridge the 0.5-meter distance gap between them.
Using the formula:
s = ut
where:
s = distance
u = velocity
t = time
For the car:
s = 0.5 meters
u = 2.5 m/s (velocity of the car after accelerating)
t = time (we need to find this)
Rearranging the formula:
t = s / u
t = 0.5 / 2.5
t = 0.2 seconds
So it takes the car an additional 0.2 seconds to catch up to the truck.
Step 4: Determine the total time for the car to pass the truck.
Total time = 2 seconds (for the car to reach 2.5 m/s) + 0.2 seconds (for the car to catch up to the truck)
Total time = 2.2 seconds
Step 5: Calculate the distance covered by the car and the truck in 2.2 seconds.
Using the formula:
s = ut
where:
s = distance
u = velocity
t = time
For the car:
u = 2.5 m/s (velocity of the car after accelerating)
t = 2.2 seconds
Substituting the values:
s_car = 2.5 * 2.2
s_car = 5.5 meters
For the truck:
u = 1.5 m/s (constant velocity)
t = 2.2 seconds
Substituting the values:
s_truck = 1.5 * 2.2
s_truck = 3.3 meters
Therefore, in 2.2 seconds, the car travels 5.5 meters while the truck travels 3.3 meters.
Step 6: Calculate the total distance from the traffic light.
Total distance = distance covered by the car + distance covered by the truck
Total distance = 5.5 + 3.3
Total distance = 8.8 meters
Converting meters to kilometers:
Total distance = 8.8 meters / 1000
Total distance = 0.0088 kilometers
Therefore, the car passes the truck in 2.2 seconds, and they have gone 0.0088 kilometers from the traffic light at that time.