Sitting besides a friend on a park bench, you grab her hat and start running in a straight line away from her. Over the first 15.0, you accelerate at 1 m/s^2 up to your maximum running speed. You then continue at your maximum running speed for 15 s more before your friend catches you. Calculate how far from the bench did you get before being caught and how long did it took your friend to catch up with you.
v2 = sqrt((v1)^2 +2a(x1)) = 5.48m/s
x2 = x1 + (v2 * t2) = 97m
dt = t1 + t2
so, t1 = (v1 - v2)/a = 5.48 s
d1 = 0.5a*t^2 = 15 m.
0.5*t^2 = 15
t^2 = 30
T1 = 5.477 s.
V = a*t = 1m/s^2 * 5.477s = 5.477 m/s.
d2 = V * t = 5.477m/s * 15s = 82.2 m.
D = d1 + d2 = 15 + 82.2 = 97.2 m. = Distance at which he was caught.
T = T1 + T2 = 5.477 + 15 = 20.48 s To
catch up.
To solve this problem, we will calculate the distance you traveled before being caught and the time it took for your friend to catch up with you.
Step 1: Calculate the distance traveled during the acceleration phase.
To calculate the distance traveled during the acceleration phase, we can use the formula:
d = (1/2) * a * t^2
where:
d = distance traveled
a = acceleration
t = time
In this case, the acceleration is 1 m/s^2 and the time is 15.0 s.
Plugging in the values:
d = (1/2) * 1 * (15.0)^2
d = 1/2 * 1 * 225
d = 112.5 meters
So, during the acceleration phase, you traveled 112.5 meters.
Step 2: Calculate the distance traveled during the constant speed phase.
To calculate the distance traveled during the constant speed phase, we can use the formula:
d = v * t
where:
d = distance traveled
v = speed or velocity
t = time
In this case, the speed is constant for 15 s.
Plugging in the values:
d = v * t
d = v * 15
112.5 = v * 15
v = 112.5 / 15
v = 7.5 m/s
So, your maximum running speed is 7.5 m/s.
Step 3: Calculate the total distance traveled.
To calculate the total distance traveled, we add the distances traveled during the acceleration phase and the constant speed phase.
Total distance = distance during acceleration + distance during constant speed
Total distance = 112.5 + (7.5 * 15)
Total distance = 112.5 + 112.5
Total distance = 225 meters
So, you traveled a total distance of 225 meters before being caught by your friend.
Step 4: Calculate the time taken by your friend to catch up with you.
To calculate the time taken by your friend to catch up with you, we can use the formula:
t = d / v
where:
t = time
d = distance traveled (225 meters)
v = velocity (running speed)
Plugging in the values:
t = 225 / 7.5
t = 30 seconds
So, it took your friend 30 seconds to catch up with you.
Therefore, you traveled a distance of 225 meters before being caught, and it took your friend 30 seconds to catch up with you.
To solve this problem, we can break it down into two parts: the time it takes for you to reach your maximum running speed and the distance you cover while running at your maximum speed.
Let's first calculate the time it takes for you to reach your maximum running speed. We know that over the first 15.0 s, you accelerate at 1 m/s^2. We can use the kinematic equation:
v = u + at
Where:
v is the final velocity
u is the initial velocity
a is the acceleration
t is the time
Initially, your velocity is 0 m/s since you start from rest. After 15.0 s, your velocity will reach its maximum value. We can calculate it as follows:
v = 0 + (1 m/s^2)(15.0 s)
v = 15.0 m/s
Now that we know the maximum velocity, we can calculate the distance covered during the acceleration phase. We can use the kinematic equation:
s = ut + (1/2)at^2
Where:
s is the distance
u is the initial velocity
a is the acceleration
t is the time
For this case, the initial velocity (u) is 0 m/s, the acceleration (a) is 1 m/s^2, and the time (t) is 15.0 s. We substitute these values into the equation:
s = (0 m/s)(15.0 s) + (1/2)(1 m/s^2)(15.0 s)^2
s = 0 m + (1/2)(1 m/s^2)(225.0 s^2)
s = 0 m + (1/2)(225.0 m)
s = 0 m + 112.5 m
s = 112.5 m
Therefore, during the acceleration phase, you cover a distance of 112.5 meters.
Next, we can calculate the distance covered while running at your maximum speed for 15.0 seconds. Since your maximum speed is reached during the acceleration phase and maintained afterward, your maximum velocity is 15.0 m/s.
To find the distance covered during this time, we can use the formula:
s = vt
Where:
s is the distance
v is the velocity
t is the time
Substituting the values:
s = (15.0 m/s)(15.0 s)
s = 225.0 m
Therefore, during the constant speed phase, you cover a distance of 225.0 meters.
Finally, to find the total distance covered before being caught, we sum the distances covered during the acceleration and constant speed phases:
Total distance = distance during acceleration + distance during constant speed
Total distance = 112.5 m + 225.0 m
Total distance = 337.5 m
Therefore, you get caught at a distance of 337.5 meters from the bench.
To find how long it took your friend to catch up with you, we need to determine the time it took for your friend to cover the same 337.5 meters. Since your friend was at rest initially and then catches up to you, their velocity will be the same during the entire time.
Using the formula:
s = vt
Where:
s is the distance
v is the velocity
t is the time
We can rearrange the equation to solve for time:
t = s / v
Substituting the values:
t = 337.5 m / 15.0 m/s
t = 22.5 s
Therefore, it took your friend 22.5 seconds to catch up with you.