a man run, accelerating at 0.5 m/s^2, for 8 sec. He then slides to a stop in 1.5 sec.
a.) What total distance did he travel?
b.) what was his average velocity?
a. d1 = 0.5a*t^2 = 0.25*8^2 = 16 m.
V1 = at = 0.5 * 8=4 m/s=Max. velocity.
a = (V-V1)/t = (0-4) / 1.5=-2.67m/s^2.
d2 = (V^2-V1^2)/2a.
d2=(0-16) / -5.33=3 m.=Sliding distance.
Dt = d1 + d2 = 16 + 3 = 19 m. = Total
dist. traveled.
b. Vavg = Dt/T = 19 / (8+1.5) = 2m/s.
To find the total distance traveled by the man, we need to break down his motion into two parts: the time interval when he was accelerating and the time interval when he was decelerating.
Step 1: Determine the distance during acceleration.
To find the distance traveled during the acceleration phase, we can use the equation:
distance = initial velocity * time + (1/2) * acceleration * time^2
In this case, the initial velocity is 0 m/s (since the man started from rest), the acceleration is 0.5 m/s^2, and the time is 8 seconds. Plugging these values into the equation, we get:
distance = 0 * 8 + (1/2) * 0.5 * (8^2)
Calculating this, we find that the distance during acceleration is 16 meters.
Step 2: Determine the distance during deceleration.
During deceleration, the man comes to a stop. The total distance traveled during deceleration can be calculated using the equation:
distance = (final velocity^2 - initial velocity^2) / (2 * acceleration)
Here, the initial velocity is the velocity when the man was at the end of the acceleration phase, and the final velocity is 0 m/s (since he comes to a stop). The acceleration is still 0.5 m/s^2, and the time is 1.5 seconds. Plugging these values into the equation, we get:
distance = (0^2 - final velocity^2) / (2 * 0.5)
As the final velocity squared is zero, the equation simplifies to:
distance = 0 / 1
Therefore, the distance during deceleration is 0 meters.
Step 3: Calculate the total distance.
To calculate the total distance, we add the distances from the acceleration and deceleration phases together:
total distance = distance during acceleration + distance during deceleration
total distance = 16 + 0
total distance = 16 meters
Therefore, the man traveled a total distance of 16 meters.
Now, let's move on to finding the average velocity.
Average velocity is defined as the total displacement divided by the total time taken. Displacement is the change in position, which can be calculated by subtracting the initial position from the final position.
Step 1: Calculate the displacement.
During acceleration, the displacement can be determined using the equation:
displacement = initial velocity * time + (1/2) * acceleration * time^2
In this case, the initial velocity is 0 m/s, the acceleration is 0.5 m/s^2, and the time is 8 seconds. Plugging these values into the equation, we get:
displacement = 0 * 8 + (1/2) * 0.5 * (8^2)
Calculating this, we find that the displacement during acceleration is 16 meters.
Step 2: Calculate the displacement during deceleration.
Since the man comes to a stop during deceleration, the final displacement is 16 meters (equal to the displacement during acceleration).
Step 3: Calculate the total displacement.
The total displacement is the sum of the displacements during both the acceleration and deceleration phases:
total displacement = displacement during acceleration + displacement during deceleration
total displacement = 16 + 16
total displacement = 32 meters
Step 4: Calculate the average velocity.
Average velocity = total displacement / total time
The total time is the sum of the time intervals for acceleration (8 seconds) and deceleration (1.5 seconds).
total time = 8 + 1.5
total time = 9.5 seconds
Substituting the values into the equation, we get:
average velocity = 32 / 9.5
Calculating this, we find that the average velocity is approximately 3.37 m/s.
Therefore, the average velocity of the man is approximately 3.37 m/s.