A glider of length 10.1 cm moves on an air track with constant acceleration. A time interval of 0.732 s elapses between the moment when its front end passes a fixed point circled A along the track and the moment when its back end passes this point. Next, a time interval of 1.01 s elapses between the moment when the back end of the glider passes point circled A and the moment when the front end of the glider passes a second point circled B farther down the track. After that, an additional 0.336 s elapses until the back end of the glider passes point circled B.
(b) Find the acceleration of the glider.
(c) Explain how you can compute the acceleration without knowing the distance between points circled A and circled B.
To find the acceleration of the glider, we can use the equations of motion.
Let's start by calculating the initial velocity (u) of the glider when it passes point A.
Using the first time interval, t₁ = 0.732 s, and the length of the glider, L = 10.1 cm = 0.101 m, we can calculate the initial velocity using the equation:
u = (2 * L) / t₁
Substituting the values, we get:
u = (2 * 0.101 m) / 0.732 s
Simplifying gives:
u = 0.276 m/s
Now let's calculate the final velocity (v) of the glider when it passes point B.
Using the second time interval, t₂ = 1.01 s, and the length of the glider, L = 0.101 m, we can calculate the final velocity using the equation:
v = (2 * L) / t₂
Substituting the values, we get:
v = (2 * 0.101 m) / 1.01 s
Simplifying gives:
v = 0.2 m/s
Now we can calculate the average acceleration (a) of the glider using the equation:
a = (v - u) / t
where t is the total time interval for the glider to travel from point A to point B, which is the sum of the second and third time intervals:
t = t₂ + t₃
Substituting the values, we get:
t = 1.01 s + 0.336 s
t = 1.346 s
Now, substituting all the known values into the acceleration equation, we get:
a = (0.2 m/s - 0.276 m/s) / 1.346 s
Simplifying gives:
a = -0.056 m/s²
Therefore, the acceleration of the glider is approximately -0.056 m/s².
To compute the acceleration without knowing the distance between points circled A and circled B, we can use the concept of average velocity. The average velocity (v_avg) of an object is given by the formula:
v_avg = (s₂ - s₁) / t
where s₁ and s₂ are the positions of the object at two different times, and t is the time taken to move between those two positions.
Since we don't know the distances s₁ and s₂, we can express them in terms of the length of the glider, L:
s₁ = 0
s₂ = L
Using this information, we can rewrite the formula for average velocity as:
v_avg = L / t
Now, since the average velocity is defined as the change in position divided by the time taken, and acceleration is defined as the change in velocity divided by the time taken, we can equate these two expressions:
v_avg = a_avg = L / t
Thus, the average velocity of the glider is equal to the average acceleration. Therefore, by calculating the average velocity using the times taken for the different intervals, we can determine the acceleration of the glider without knowing the distance between points A and B.