A man ruins at a velocity of 4.5 m/s for 15 minutes. When
going up an increasingly steep hill, he slows down at a constant rate of 0.05 m/? for 90.0 seconds and comes to a stop. How far did he run?
To find the distance the man ran, we need to find the total distance traveled during the initial velocity and the distance traveled during the deceleration.
The initial velocity is 4.5 m/s, and the time for which he ran at this velocity is 15 minutes. We need to convert minutes to seconds:
15 minutes * 60 seconds/minute = 900 seconds
The distance traveled during the initial velocity can be calculated using the formula:
distance = velocity * time
distance = 4.5 m/s * 900 seconds = 4050 meters
Now let's find the distance traveled during the deceleration. The deceleration rate is -0.05 m/s², and the time for which he decelerates is 90.0 seconds. We need to calculate the final velocity using the formula:
final velocity = initial velocity + (acceleration * time)
final velocity = 4.5 m/s + (-0.05 m/s² * 90.0 seconds) = 4.5 m/s - 4.5 m/s = 0 m/s
Now we can calculate the distance traveled during deceleration using the formula:
distance = (initial velocity + final velocity) / 2 * time
distance = (4.5 m/s + 0 m/s) / 2 * 90.0 seconds = 2.25 m/s * 90.0 seconds = 202.5 meters
The total distance traveled is the sum of the distance during the initial velocity and the distance during deceleration:
total distance = 4050 meters + 202.5 meters = 4252.5 meters
Therefore, the man ran a total distance of 4252.5 meters.
To find the distance the man ran, we need to calculate the distance he covered before coming to a stop.
1. First, convert the time taken to run from minutes to seconds:
15 minutes = 15 * 60 = 900 seconds
2. Next, calculate the distance covered with constant velocity using the formula:
distance = velocity * time
distance = 4.5 m/s * 900 s = 4050 meters
3. In the next step, we need to find the time it took for the man to come to a stop. The rate of slowing down is given as 0.05 m/s per second, and the time given is 90.0 seconds. However, the distance covered is not given, so we'll need to calculate it.
4. Using the formula of motion with uniform acceleration:
final velocity = initial velocity + (acceleration * time)
0 m/s = 4.5 m/s + (-0.05 m/s^2 * 90 s)
5. Rearranging the equation to solve for the initial velocity:
initial velocity = - (acceleration * time)
Plugging in the values, we get:
initial velocity = - (0.05 m/s^2 * 90 s) = -4.5 m/s
6. With the initial and final velocities, we can find the distance covered using the formula:
distance = (final velocity^2 - initial velocity^2) / (2 * acceleration)
Plugging in the values, we get:
distance = (0^2 - (-4.5 m/s)^2) / (2 * -0.05 m/s^2)
distance = (-20.25 m^2/s^2) / (-0.1 m/s^2) = 202.5 meters
7. Finally, we can find the total distance covered by adding the distances covered with constant velocity and during deceleration:
total distance = distance with constant velocity + distance during deceleration
total distance = 4050 meters + 202.5 meters = 4252.5 meters
Therefore, the man ran a total distance of 4252.5 meters.