A truck traveling at a constant speed of 40.0 km/h applies its brakes and comes to a complete stop in 5.0 s. The truck starts again and accelerates at a constant rate of 0.80 m/s2. After restarting, how far does the truck travel before reaching its original speed of 40.0 km/h?
I am not certain what How far means? The total distance, or just the distance accelearting?
If the distance accelerating
Vf^2=2ad
change km/hr to m/s, solve for d.
To find the distance the truck travels before reaching its original speed of 40.0 km/h, we need to divide the problem into two parts: the time it took to stop and the time it took to accelerate back to the original speed.
1. Time to stop:
We are given:
Initial speed (u) = 40.0 km/h
Final speed (v) = 0 km/h
Time (t) = 5.0 s
Using the formula v = u + at, where a is the acceleration and t is the time, we can solve for the acceleration:
0 km/h = 40.0 km/h + a * 5.0 s
Converting the speeds to m/s:
0 m/s = 40.0 km/h * (1000 m/1 km) * (1 h/3600 s) + a * 5.0 s
Simplifying the equation:
0 m/s = (40000 m/3600 s) + a * 5.0 s
0 m/s = 11.11 m/s + 5a
Now, we can solve for the acceleration:
5a = -11.11 m/s
a = -11.11 m/s / 5
a = -2.22 m/s^2
Now, we can use the equation d = ut + 0.5at^2, where d is the distance, and u is the initial velocity:
d = 40.0 km/h * (1000 m/1 km) * (1 h/3600 s) * 5.0 s + 0.5 * (-2.22 m/s^2) * (5.0 s)^2
Simplifying the equation:
d = 11.11 m/s * 5.0 s - 0.5 * 2.22 m/s^2 * (5.0 s)^2
Calculating:
d = 55.55 m - 27.77 m
d ≈ 27.78 m
Therefore, the truck travels approximately 27.78 meters before coming to a complete stop.
2. Time to accelerate back to the original speed:
Acceleration (a) = 0.80 m/s^2
Final speed (v) = 40.0 km/h
Initial speed (u) = 0 m/s
Using the formula v = u + at, we can solve for the time (t):
40.0 km/h = 0 m/s + 0.80 m/s^2 * t
Converting the speed to m/s:
40.0 km/h = 0.80 m/s^2 * t
40.0 km/h * (1000 m/1 km) * (1 h/3600 s) = 0.80 m/s^2 * t
Simplifying the equation:
11.11 m/s = 0.80 m/s^2 * t
Now, we can solve for the time (t):
t = 11.11 m/s / 0.80 m/s^2
t ≈ 13.89 s
Finally, we can calculate the distance traveled during acceleration:
Using the equation d = ut + 0.5at^2:
d = 0 m/s * 13.89 s + 0.5 * 0.80 m/s^2 * (13.89 s)^2
Simplifying the equation:
d = 0 + 0.5 * 0.80 m/s^2 * (13.89 s)^2
Calculating:
d = 0 + 0.5 * 0.80 m/s^2 * 192.8 s^2
d ≈ 77.12 m
Therefore, the truck travels approximately 77.12 meters during acceleration.
To find the total distance traveled before reaching its original speed, we add the distances from stopping and acceleration:
Total distance = Distance during stopping + Distance during acceleration
Total distance = 27.78 m + 77.12 m
Total distance ≈ 104.9 m
Therefore, the truck travels approximately 104.9 meters before reaching its original speed of 40.0 km/h.
To find the distance the truck travels before reaching its original speed, we need to break down the problem into two parts:
1. Stopping distance: First, we need to calculate how far the truck travels when it comes to a complete stop. We can use the equation of motion:
v = u + at,
where v is the final velocity (0 m/s), u is the initial velocity (40.0 km/h converted to m/s), a is the acceleration (negative since it's decelerating), and t is the time taken to stop (5.0 s).
Converting 40.0 km/h to m/s:
40.0 km/h = 40.0 * (1000/3600) = 11.1 m/s.
Plugging the values into the equation:
0 = 11.1 + (-a) * 5.0,
-11.1 = -5.0a,
a = 11.1/5.0,
a ≈ 2.22 m/s^2.
Now, we have the acceleration (a), initial velocity (u), and time (t), so we can use the equation:
s = ut + (1/2)at^2,
where s is the distance traveled.
Plugging in the values:
s = 11.1 * 5.0 + (0.5) * 2.22 * (5.0)^2,
s ≈ 27.75 + 27.75,
s ≈ 55.50 meters.
Therefore, the truck travels approximately 55.50 meters before coming to a complete stop.
2. Distance while accelerating: Next, we need to calculate the distance the truck travels while accelerating to reach its original speed.
Here, we can use the equation of motion:
v^2 = u^2 + 2as,
where v is the final velocity (40.0 km/h converted to m/s), u is the initial velocity (0 m/s), a is the acceleration (0.80 m/s^2), and s is the distance.
Converting 40.0 km/h to m/s:
40.0 km/h = 40.0 * (1000/3600) = 11.1 m/s.
Plugging the values into the equation:
(11.1)^2 = 0^2 + 2 * 0.80 * s,
123.21 = 1.60s,
s = 123.21/1.60,
s ≈ 77.00 meters.
Therefore, the truck travels approximately 77.00 meters while accelerating.
Finally, to find the total distance traveled, we add the stopping distance and the distance while accelerating:
Total distance = Stopping distance + Distance while accelerating,
Total distance ≈ 55.50 + 77.00,
Total distance ≈ 132.50 meters.
Hence, the truck travels approximately 132.50 meters before reaching its original speed of 40.0 km/h.