1. A car is traveling down the road at a constant speed of 11 km/h. At time zero it is forced to stop. The driver slams on the brakes and the car travels 18 m as it uniformly accelerates, coming to a stop. The car remains at rest for 5.0 seconds. Then the driver shifts into reverse and begins to back up. It accelerates from rest to a speed of 6.0 m/s in 4.0 seconds. The car then travels at this speed for 5.0 seconds. (a) Make a velocity vs time graph for the car’s motion. (b) At the end of all this motion, what is the car’s position?
To solve this problem, we will break it down into several steps.
Step 1: Find the acceleration during the first phase of motion when the car is forced to stop.
Given:
Initial velocity, u = 11 km/h = (11 * 1000 m) / (60 * 60 s) = 3.06 m/s
Final velocity, v = 0 m/s
Displacement, s = 18 m
Using the formula for uniformly accelerated motion:
v^2 = u^2 + 2as
0^2 = (3.06)^2 + 2a(18)
Solving for acceleration (a):
a = (-2(3.06)^2) / (2 * 18)
Step 2: Find the time taken to accelerate to a speed of 6.0 m/s during the second phase of motion.
Initial velocity, u = 0 m/s
Final velocity, v = 6.0 m/s
Time taken, t = 4.0 seconds
Using the formula for uniformly accelerated motion:
v = u + at
6.0 = 0 + a(4.0)
Solving for acceleration (a):
a = 6.0 / 4.0
Step 3: Determine the position of the car at the end of all this motion.
To calculate this, we need to find the total displacement during each phase of motion.
Phase 1: The car comes to a stop after traveling 18 m. The displacement is 18 m.
Phase 2: The car accelerates from rest to a speed of 6.0 m/s. The displacement can be calculated using the formula:
s = ut + (1/2)at^2
For the first 4 seconds:
s = (1/2)(0)(4)^2 = 0
For the next 5 seconds:
s = (1/2)(6.0)(5)^2 = 75.0 m
Therefore, the total displacement during Phase 2 is 75.0 m.
Lastly, we need to determine the final position of the car after it traveled 18 m, remained at rest for 5.0 seconds, and then traveled 75.0 m.
To do this, we add up the total displacement from each phase:
Total displacement = displacement during Phase 1 + displacement during Phase 2
Total displacement = 18 m + 75.0 m
Total displacement = 93.0 m
Therefore, at the end of all this motion, the car’s position is 93.0 meters.
(a) To create a velocity-time graph, plot the following points:
For Phase 1: (0 sec, 3.06 m/s) and (5 sec, 0 m/s)
For Phase 2: (5 sec, 0 m/s) and (9 sec, 6.0 m/s)
Connect the points with straight lines to create the velocity vs. time graph.
(b) The car's position at the end of all motion is 93.0 meters.