A car is stopped at a set of traffic lights. It takes off, accelerating at 4.5m/s2 until it is travelling at 60km/h. It maintains this velocity for 3 minutes before braking to a halt over 60m at another set of traffic lights. How far has the car travelled in this trip and how long did the trip take? Plot a velocity vs time graph for this situation.
V = 60km/h = 60000m/3600s = 16.7 m/s.
a = (V-Vo)/t = 4.5 m/s^2
(16.7-0)/t = 4.5
t = 16.7/4.5 = 3.70 s.
d1 = 0.5a*t^2 = 2.25*3.7^2 = 31 m.
d2 = V*t = 16.7m/s * 180s = 3,000 m.
d3 = 60 m.
d1+d2+d3 = 30.8 + 3000 + 60 = 3091 m. =
Total distance traveled.
V^2 = Vo^2 + 2a*d
a = (V^2-Vo^2)/2d =(0-16.7^2)/120=-2.32m/s^2.
V = Vo - 2.32t = 0
16.7 - 2.32t = 0
2.32t = 16.7
t = 7.2 s. To stop.
T = 3.70 + 180 + 7.2 = 190.9 s. = Total
travel time.
To find the distance traveled by the car during the trip, we need to calculate the distance traveled during acceleration, the distance traveled during constant velocity, and the distance traveled during deceleration.
1. Distance during acceleration:
We can use the equation of motion:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
We are given the initial velocity as zero, acceleration as 4.5 m/s^2, and the final velocity as 60 km/h. First, we need to convert the final velocity to m/s:
60 km/h = (60 * 1000) / (60 * 60) = 16.67 m/s
Substituting the values, we have:
distance1 = (0 * t1) + (0.5 * 4.5 * t1^2)
2. Distance during constant velocity:
The car travels at a constant velocity of 60 km/h for 3 minutes. We need to convert this to seconds:
3 minutes = 3 * 60 = 180 seconds
Since the velocity is constant, the distance traveled during constant velocity is given by:
distance2 = velocity * time
distance2 = 16.67 m/s * 180 s
3. Distance during deceleration:
The car comes to a halt over a distance of 60 m. This distance is the sum of the distances traveled during deceleration:
distance3 = 60 m
Now, we can calculate the total distance traveled by adding up the distances:
Total distance = distance1 + distance2 + distance3
To find the time taken for the trip, we need to add up the time during acceleration, constant velocity, and deceleration.
1. Time during acceleration:
Using the equation of motion:
final velocity = initial velocity + (acceleration * time)
Substituting the values, we have:
16.67 m/s = 0 + (4.5 * t1)
2. Time during constant velocity:
We already know that the time for this phase is 180 seconds.
3. Time during deceleration:
We can use the equation of motion:
final velocity = initial velocity + (acceleration * time)
Substituting the values, we have:
0 = 16.67 m/s + (-6.67 * t3)
Adding the three times will give us the total trip duration.
Now let's plot a velocity vs time graph for this situation. The graph will have three different regions: an increasing velocity during acceleration, a constant velocity during constant velocity, and a decreasing velocity during deceleration.