A glider of length 12.7 cm moves on an air track with constant acceleration. A time interval of 0.563 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 1.11 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 0.499 s elapses until the back end of the glider passes point circled B.
Find the acceleration of the glider.
To find the acceleration of the glider, we can use the equations of motion. The key equation we need to use is:
\[s = ut + \frac{1}{2}at^2\]
where:
- \(s\) is the displacement
- \(u\) is the initial velocity
- \(a\) is the acceleration
- \(t\) is the time
We can find the displacement and time values for each part of the glider's motion and use them to calculate the acceleration.
1. First, let's analyze the motion between points A and B.
- The time interval between the front and back end passing point A is 0.563 s. Since the glider is moving with a constant acceleration, the average velocity is equal to the final velocity.
- The displacement between the front and back end is the length of the glider, which is 12.7 cm.
Using the equation of motion, we have:
\[s_{AB} = 12.7 \, \text{cm} = \frac{1}{2} \cdot a_{AB} \cdot (0.563 \, \text{s})^2\]
2. Next, let's analyze the motion between points B and back end passing point B.
- The time interval between the back end passing point A and the front end passing point B is 1.11 s. Again, the average velocity is equal to the final velocity.
- The displacement between the back end and front end is again the length of the glider, which is 12.7 cm.
Using the equation of motion, we have:
\[s_{B} = 12.7 \, \text{cm} = \frac{1}{2} \cdot a_{B} \cdot (1.11 \, \text{s})^2\]
3. Finally, let's analyze the motion between points A and back end passing point B.
- The time interval between the back end passing point A and the back end passing point B is 0.499 s.
- The displacement is the entire length of the glider, which is 12.7 cm.
Again, using the equation of motion, we have:
\[s_{AB} = 12.7 \, \text{cm} = \frac{1}{2} \cdot a_{AB} \cdot (0.563 \, \text{s} + 1.11 \, \text{s} + 0.499 \, \text{s})^2\]
Now that we have three equations with three unknowns (accelerations), we can solve them simultaneously to find the value of the acceleration of the glider.
To find the acceleration of the glider, we can use the equations of motion:
1. Distance = Initial Velocity * Time + (1/2) * Acceleration * Time^2
We are given the following measurements:
- Length of the glider = 12.7 cm
- Time interval between the front and back end passing Point A = 0.563 s
- Time interval between the back end passing Point A and the front end passing Point B = 1.11 s
- Time interval between the front end passing Point B and the back end passing Point B = 0.499 s
Let's break down the problem into steps:
Step 1: Find the initial velocity of the glider
Since the glider is at rest before it starts moving, the initial velocity is 0.
Step 2: Find the distance traveled by the glider between the front and back end passing Point A
Using the formula Distance = Initial Velocity * Time + (1/2) * Acceleration * Time^2, we can substitute the given values:
12.7 cm = 0 * 0.563 s + (1/2) * Acceleration * (0.563 s)^2
Simplifying the equation:
12.7 cm = 0.158 s^2 * Acceleration
Step 3: Find the distance traveled by the glider between the back end passing Point A and the front end passing Point B
Similarly, we can use the same formula since the glider maintains the same acceleration:
12.7 cm = 0 * 1.11 s + (1/2) * Acceleration * (1.11 s)^2
Simplifying the equation:
12.7 cm = 0.617 s^2 * Acceleration
Step 4: Find the distance traveled by the glider between the front end passing Point B and the back end passing Point B
Again, we use the same formula:
12.7 cm = 0.563 s * Initial Velocity + (1/2) * Acceleration * (0.563 s)^2
Simplifying the equation:
12.7 cm = 0.563 s * 0 + 0.079 s^2 * Acceleration
Step 5: Combine the equations to solve for the acceleration
We have three equations with the same unknown, Acceleration. We can set up a system of equations to solve for it:
0.158 s^2 * Acceleration = 12.7 cm
0.617 s^2 * Acceleration = 12.7 cm
0.079 s^2 * Acceleration = 12.7 cm
To simplify the calculations, we convert cm to m by dividing both sides of the equations by 100:
0.00158 s^2 * Acceleration = 0.127 m
0.00617 s^2 * Acceleration = 0.127 m
0.00079 s^2 * Acceleration = 0.127 m
Now we have a system of equations:
0.00158A = 0.127
0.00617A = 0.127
0.00079A = 0.127
Solving for A, we get:
A = 0.127 / 0.00158
A = 80.38 m/s^2 (approximately)
Therefore, the acceleration of the glider is approximately 80.38 m/s^2.