A cyclist starts from rest and accelerates at 2m/s2, for 2second. Her velocity remains constant for the following 4second. The brakes are then applied and the bicycle decelerates uniformly until it comes to rest after 10 second.
Find(a) the distance moved during each stage of motion.
(b) the average velocity over the whole journey.
V = Vo + a*t = 0 + 2*2 = 4 m/s.
a. d1 = v*t = 4 * 4 = 16 m.
V = Vo + a*t = 0, 4 + a*10 = 0,
a = -0.4 m/s^2.
V^2 = Vo^2 + 2a*d2 = 0, 4^2 - 0.8*d = 0, -0.8d2 = -16, d2 = 20 m.
b. Vavg = (d1+d2)/(t1+t2) = (16+20)/(4+10) = 2.57 m/s.
A cyclist speeds up at a constant acceleration of 2m/s? from rest. Find the velocity of the cyclist after 10 second.
2m
(a) Stage 1: Acceleration for 2 seconds
Using the formula for distance traveled during acceleration:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Initial velocity = 0 (since the cyclist starts from rest)
Acceleration = 2 m/s^2
Time = 2 seconds
distance = (0 * 2) + (0.5 * 2 * 2^2)
distance = 0 + (0.5 * 2 * 4)
distance = 0 + 4
distance = 4 meters
Stage 2: Constant velocity for 4 seconds
Since velocity remains constant, distance = velocity * time
Velocity = 2 m/s (the same as the final velocity in the previous stage)
Time = 4 seconds
distance = 2 * 4
distance = 8 meters
Stage 3: Deceleration until coming to rest
The final velocity is 0 (since the bicycle comes to rest)
Again using the formula for distance traveled during deceleration:
distance = (initial velocity * time) - (0.5 * acceleration * time^2)
Initial velocity = 2 m/s (the same as the final velocity in the previous stage)
Acceleration = ? (We need to figure this out)
Time = 10 seconds
We can solve for acceleration using the formula:
0 = (2 * 10) - (0.5 * acceleration * 10^2)
0 = 20 - (0.5 * acceleration * 100)
0 = 20 - 50 * acceleration
50 * acceleration = 20
acceleration = 20 / 50
acceleration = 0.4 m/s^2
Now we can find the distance:
distance = (2 * 10) - (0.5 * 0.4 * 10^2)
distance = 20 - (0.5 * 0.4 * 100)
distance = 20 - (0.2 * 100)
distance = 20 - 20
distance = 0 meters
(b) The average velocity over the whole journey is given by:
total distance / total time
Total distance = distance in stage 1 + distance in stage 2 + distance in stage 3
Total distance = 4 + 8 + 0
Total distance = 12 meters
Total time = time in stage 1 + time in stage 2 + time in stage 3
Total time = 2 + 4 + 10
Total time = 16 seconds
Average velocity = 12 meters / 16 seconds
Average velocity = 0.75 m/s
So, the average velocity over the whole journey is 0.75 m/s.
To find the distance moved during each stage of motion, we can use the kinematic equations of motion.
(a) Distance moved during each stage of motion:
1. Acceleration phase:
Given:
Initial velocity (u) = 0 m/s (rest)
Acceleration (a) = 2 m/s^2
Time (t) = 2 s
Using the equation of motion:
s = ut + (1/2)at^2
Substituting the given values:
s = 0 + (1/2)(2)(2)^2
s = 0 + 2(4)
s = 8 meters
Therefore, the distance moved during the acceleration phase is 8 meters.
2. Constant velocity phase:
Given:
Velocity (v) = constant
Time (t) = 4 s
Since the velocity remains constant, the distance moved can be calculated using the equation:
s = vt
Substituting the given values:
s = v(4)
Since the velocity is constant, the distance moved during this phase depends on the average velocity over the entire journey, which we will calculate later.
3. Deceleration phase:
Given:
Final velocity (v) = 0 m/s (comes to rest)
Time (t) = 10 s
Using the same equation of motion:
s = vt + (1/2)at^2
Substituting the given values:
s = 0(10) + (1/2)a(10)^2
s = 0 + (1/2)(-a)(100)
s = -50a
Since the deceleration is uniform, we can substitute "a" with the absolute value of the acceleration (2 m/s^2) to get a positive distance.
s = -50(2)
s = -100 meters
Therefore, the distance moved during the deceleration phase is 100 meters.
(b) Average velocity over the whole journey:
To calculate the average velocity over the entire journey, we need the total displacement (change in position) and the total time taken.
Total displacement = distance moved during acceleration phase + distance moved during constant velocity phase + distance moved during deceleration phase
Total displacement = 8 meters + s (distance moved during constant velocity phase) - 100 meters
Total time taken = time taken during acceleration phase + time taken during constant velocity phase + time taken during deceleration phase
Total time taken = 2 seconds + 4 seconds + 10 seconds
Average velocity = Total displacement / Total time taken
Average velocity = (8 + s - 100) / (2 + 4 + 10)
Since the velocity remains constant during the constant velocity phase, we can say s = vt, where v is the velocity during the constant velocity phase. Thus, s = v(4).
Average velocity = (8 + v(4) - 100) / 16
Now, substitute v with the value obtained from the constant velocity phase, and calculate the final average velocity.