An electric motor is used to pull a 125 kg box across a floor using a long cable. the tension in the cable is 350N and the box accelerates at 1.2m/s^2[forward] for 5.0s. The cable breaks and the box slows down and stops.
How far does the box travel from the moment the cable breaks until it stops?
To find the distance traveled by the box after the cable breaks until it stops, we need to use the equations of linear motion. We'll start by finding the initial velocity of the box before the cable breaks.
Given:
Mass of the box (m) = 125 kg
Acceleration (a) = 1.2 m/s^2 (forward)
Time (t) = 5.0 s
Using the equation:
v = u + at
Where:
v = final velocity
u = initial velocity
a = acceleration
t = time
Here, the box starts from rest, so the initial velocity (u) is 0 m/s.
v = u + at
v = 0 + (1.2 m/s^2)(5.0 s)
v = 6.0 m/s
The final velocity of the box right before it stops is 6.0 m/s.
Now, to calculate the distance traveled by the box, we can use the equation:
s = ut + (1/2)at^2
Where:
s = distance
u = initial velocity
t = time
a = acceleration
Since the box starts from rest, the initial velocity (u) is 0 m/s.
s = (0 m/s)(5.0 s) + (1/2)(1.2 m/s^2)(5.0 s)^2
s = 0 + (0.5)(1.2 m/s^2)(25 s^2)
s = 0.5(1.2 m/s^2)(25 s^2)
s = 15 m
The box travels a distance of 15 meters from the moment the cable breaks until it stops.
To find the distance traveled by the box from the moment the cable breaks until it comes to a stop, we can use the equation of motion:
distance = initial velocity * time + (1/2) * acceleration * time^2.
First, let's find the initial velocity of the box. We know that the tension in the cable is equal to the force acting on the box, given by Newton's second law: force = mass * acceleration.
Therefore, 350N = 125kg * 1.2m/s^2.
Rearranging the equation, we find:
acceleration = force / mass.
1.2m/s^2 = 350N / 125kg = 2.8m/s^2.
Now we have the acceleration of the box. We can use this value to calculate the initial velocity:
initial velocity = final velocity - (acceleration * time).
Since the box comes to a stop, the final velocity is zero. Thus:
initial velocity = 0 - (2.8m/s^2 * 5.0s).
initial velocity = -14m/s.
The negative sign indicates that the box was moving in the opposite direction before it came to a stop. However, we only need the magnitude of the initial velocity for calculating the distance traveled, so we can ignore the negative sign.
Now we can substitute the values into the equation of motion to find the distance:
distance = (-14m/s * 5s) + (1/2 * 2.8m/s^2 * (5s)^2).
distance = (-70m) + (1/2 * 2.8m/s^2 * 25s^2).
distance = (-70m) + (1.4m/s^2 * 25s^2).
distance = (-70m) + (1.4m/s^2 * 625s^2).
distance = (-70m) + (875m).
distance = 805m.
Therefore, the box travels a distance of 805 meters from the moment the cable breaks until it stops.