Final answer:
To determine the minimum constant acceleration required for the motorcycle to safely cross the canyon, we can analyze the motion of the motorcycle. The minimum constant acceleration in the x-direction must be 40 m/s².
Step-by-step explanation:
To determine the minimum constant acceleration required for the motorcycle to safely cross the canyon, we need to analyze the motion of the motorcycle. Since the motorcycle is fired horizontally, we only need to consider the motion in the x-direction. The initial velocity in the x-direction is 0 m/s and the final velocity is the velocity required to cover the horizontal distance of 1.20 km (1200 m) in the given time. Using the equation v = u + at, we can solve for the acceleration:
u = 0 m/s (initial velocity in x-direction)
v = ? (final velocity in x-direction)
t = 80 s (time to cover the first 2 km)
Using v = u + at, we have v = 0 + a(80), which simplifies to:
v = 80a
The final velocity is the velocity required to cover the remaining horizontal distance of 1.20 km (1200 m) in the given time:
u = 0 m/s (initial velocity in x-direction)
v = ? (final velocity in x-direction)
t = ? (time needed to cover 1.20 km)
Since the total time to cross the canyon is 120 seconds, the time to cover the remaining distance is 120 - 80 = 40 seconds. Using v = u + at, we have v = 0 + a(40), which simplifies to:
v = 40a
Since the initial velocity and final velocity for the entire motion in the x-direction are the same, we can equate the two expressions for v:
80a = 40a
Dividing both sides by 40a, we get:
80 = 40
So, the minimum constant acceleration in the x-direction that must be provided by the engines for the cycle to cross the canyon safely is 40 m/s².