A shuffleboard disk is accelerated at a constant rate from rest to a speed of 4.6 m/s over a 1.4 m distance by a player using a cue. At this point the disk loses contact with the cue and slows at a constant rate of 2.6 m/s2 until it stops.
(a) How much time elapses from when the disk begins to accelerate until it stops?
(b) What total distance does the disk travel?
To solve this problem, we can use the basic kinematic equations of motion.
(a) To find the time elapsed from when the disk begins to accelerate until it stops, we need to find the time it takes for the disk to reach its maximum speed and the time it takes for the disk to decelerate to a stop.
First, let's find the time it takes for the disk to reach its maximum speed using the equation:
v = u + at
where:
v = final velocity (4.6 m/s)
u = initial velocity (0 m/s, as the disk starts from rest)
a = acceleration (unknown)
t = time
Substituting the given values into the equation, we get:
4.6 = 0 + a * t
Next, to find the time it takes for the disk to decelerate to a stop, we can use the equation:
v² = u² + 2as
where:
v = final velocity (0 m/s)
u = initial velocity (4.6 m/s, as the disk is moving at this speed before it starts to slow down)
a = acceleration (constant deceleration of 2.6 m/s²)
s = distance traveled during the deceleration (unknown)
Substituting the given values into the equation, we get:
0 = (4.6)² + 2 * (-2.6) * s
Now, we have two equations with two unknowns (a and t). We can solve these equations simultaneously to find the values of a and t.
From the first equation:
4.6 = a * t -> (Equation 1)
Re-arranging the second equation:
0 = 21.16 - 5.2s -> 5.2s = 21.16 -> s = 4.065 m
Now, substitute the value of s in Equation 1:
0 = 4.6 - 2.6t
Simplifying,
2.6t = 4.6
t = 4.6 / 2.6
t ≈ 1.77 s
Therefore, (a) the time elapsed from when the disk begins to accelerate until it stops is approximately 1.77 seconds.
(b) To find the total distance traveled by the disk, we need to calculate the distance covered during acceleration and the distance covered during deceleration.
Distance covered during acceleration can be calculated using the equation:
s = ut + 0.5at²
where:
s = distance (1.4 m)
u = initial velocity (0 m/s)
a = acceleration (unknown)
t = time (1.77 s)
Substituting the given values into the equation, we get:
1.4 = 0 + 0.5 * a * (1.77)²
Simplifying,
1.4 = 0.5 * a * 3.1329
2.8 = a * 3.1329
a ≈ 0.892 m/s²
Now, the distance covered during deceleration can be calculated using the equation:
s = ut + 0.5at²
where:
s = distance (unknown)
u = initial velocity (4.6 m/s)
a = acceleration (-2.6 m/s²)
t = time (1.77 s)
Substituting the given values into the equation, we get:
s = 4.6 * 1.77 + 0.5 * (-2.6) * (1.77)²
s ≈ 4.891 m
Finally, the total distance traveled by the disk is the sum of the individual distances covered during acceleration and deceleration:
Total distance = Distance during acceleration + Distance during deceleration
Total distance = 1.4 m + 4.891 m
Total distance ≈ 6.291 m
Therefore, (b) the total distance traveled by the disk is approximately 6.291 meters.