Lucy is using a programmable treadmill. She inputs a top speed into the treadmill and presses "start." The treadmill
accelerates from rest at 0.100 m/s2 until it reaches that speed, after which it stays at that constant speed. Later, when she is
getting tired, she hits the "stop" button and the treadmill slows down at a rate of 0.200 m/s2 until it comes to a stop. The
treadmill tells her that she traveled the equivalent of 3.00 kilometers in 15.0 minutes while the treadmill was in motion. (a) If
Lucy had been running forward in a straight line on the ground (instead of in place on the treadmill), what would be her
average velocity in meters per second over the whole 15.0 minutes? (b) What would be her average acceleration over the
15.0 minutes? (c) What was the top speed of the treadmill? (d) How long after pressing "start" did she press "stop?"
I got a and b to be 3.3m/s and 0 m/s^2 respectively, but the next part is too hard. If you can guide me through it, it would mean a lot
Of course! Let's work through part (c) and (d) together.
To determine the top speed of the treadmill, we can use the information given in the problem. We know that the treadmill accelerates from rest at 0.100 m/s² until it reaches its top speed. Let's denote the top speed of the treadmill as "v_top."
To find the top speed, we can use one of the kinematic equations:
v = u + at
where:
v = final velocity (top speed)
u = initial velocity (0 m/s, as the treadmill starts from rest)
a = acceleration (0.100 m/s², as given in the problem)
t = time taken to reach top speed
Since the treadmill starts from rest, the initial velocity (u) is 0. We need to find the time taken to reach the top speed (t). Rearranging the equation, we have:
t = (v - u) / a
Substituting the known values:
t = (v_top - 0) / 0.100
Now, let's move on to part (d), where we have to find how long after pressing "start" Lucy pressed the "stop" button.
We know that the treadmill took 15.0 minutes to complete its motion. Let's denote the time Lucy pressed the "stop" button as "t_stop."
To find the time, we can use the kinematic equation again:
v = u + at
where:
v = final velocity (0 m/s, as the treadmill comes to a stop)
u = initial velocity (v_top, as the treadmill was at its top speed when Lucy pressed "stop")
a = acceleration (-0.200 m/s², as the treadmill slows down at this rate)
t = time taken to stop (t_stop)
Rearranging the equation:
t_stop = (v - u) / a
Substituting the known values:
t_stop = (0 - v_top) / -0.200
To solve for both top speed (v_top) and time to stop (t_stop), we need to know the value of v_top or t_stop. We'll use the given information about distance traveled to establish a relationship between the two unknowns.
The problem states that Lucy traveled the equivalent of 3.00 kilometers in 15.0 minutes while the treadmill was in motion. Since her motion is in the direction of the treadmill, her distance traveled is equivalent to the distance traveled by the treadmill belt.
We can use the equation for distance traveled:
s = ut + (1/2)at²
Given:
s = 3.00 kilometers = 3000 meters (since 1 kilometer = 1000 meters)
u = 0 m/s (initial velocity, as the treadmill starts from rest)
a = 0.100 m/s² (acceleration until reaching top speed)
We need to calculate the time it took for Lucy to reach the top speed (t) and the time it took for her to stop (t_stop).
Using the equation for distance traveled:
3000 = 0 + (1/2)(0.100)t²
Simplifying the equation:
3000 = 0.050t²
Dividing both sides by 0.050:
t² = 60000
Taking the square root of both sides:
t = √60000
Calculating the value of t, we find:
t ≈ 244.95 seconds
Now, you can substitute this value of t into the earlier equations to solve for v_top and t_stop.