A car accelerates from rest at 1.0 m/s2 for 20.0 sec along a straight road. it then moves at a constant speed for 50.0 sec. it then decelerates uniformly to a stop in 30.0 s. find the total distance covered by the car.
To find the total distance covered by the car, we need to find the distance covered during each phase of motion: acceleration, constant speed, and deceleration.
1. Acceleration phase:
The initial velocity (u) is 0 m/s, acceleration (a) is 1.0 m/s^2, and the time (t) is 20.0 s.
We can use the equation: distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Using the values given:
distance = (0 * 20) + (0.5 * 1.0 * 20^2)
distance = 0 + 0.5 * 1.0 * 400
distance = 0 + 200
distance = 200 meters
2. Constant speed phase:
The car is moving at a constant speed, which means the acceleration is 0 m/s^2.
The time for this phase is given as 50.0 s.
The distance covered during this phase can be calculated using the equation: distance = velocity * time
Since the velocity is constant, we need to determine its value.
In the previous phase, the car accelerated from rest, so its final velocity at the end of the acceleration phase will be the initial velocity for the constant speed phase.
Using the equation: final velocity = initial velocity + (acceleration * time)
final velocity = 0 + (1.0 * 20)
final velocity = 20 m/s
Using the values calculated:
distance = 20 * 50
distance = 1000 meters
3. Deceleration phase:
The final velocity is 0 m/s, the deceleration is uniform, and the time is 30.0 s.
Since the final velocity is 0, the equation for distance is similar to the one used for acceleration phase:
distance = (initial velocity * time) + (0.5 * deceleration * time^2)
Using the values given:
distance = (20 * 30) + (0.5 * deceleration * 30^2)
distance = 600 + (0.5 * deceleration * 900)
We need to find the value of the deceleration, which can be determined using the equation: final velocity = initial velocity + (acceleration * time)
0 = 20 + (deceleration * 30)
deceleration = -20/30
deceleration = -2/3 m/s^2
Substituting the value of deceleration in the distance equation:
distance = 600 + (0.5 * (-2/3) * 900)
distance = 600 - (300/3)
distance = 600 - 100
distance = 500 meters
Now, we can find the total distance covered by the car by summing up the distances from each phase:
Total distance = distance during acceleration + distance during constant speed + distance during deceleration
Total distance = 200 + 1000 + 500
Total distance = 1700 meters
Therefore, the total distance covered by the car is 1700 meters.
To find the total distance covered by the car, we need to calculate the distance covered during each phase of motion and then add them together:
1. Phase 1: Acceleration from rest at 1.0 m/s^2 for 20.0 seconds.
During this phase, we can use the equation of motion:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Since the initial velocity is zero, the equation simplifies to:
distance = 0.5 * acceleration * time^2
Plugging in the values:
distance1 = 0.5 * 1.0 m/s^2 * (20.0 s)^2
2. Phase 2: Constant speed for 50.0 seconds.
During this phase, the car is moving at a constant speed, so the distance covered is simply:
distance2 = speed * time
3. Phase 3: Deceleration to a stop in 30.0 seconds.
Since the car is decelerating uniformly, we can again use the equation of motion:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
In this case, the final velocity is zero, so the equation becomes:
distance = (initial velocity * time) - (0.5 * acceleration * time^2)
Plugging in the values:
distance3 = (constant speed * 30.0 s) - (0.5 * acceleration * (30.0 s)^2)
Finally, we can calculate the total distance covered by adding the distances from each phase:
total distance = distance1 + distance2 + distance3