An air track glider, 8.0 cm long, blocks light as it goes through a photocell gate. The glider is released from rest on a frictionless inclined track and the gate is positioned so that the glider has travelled 96 cm when it is in the middle of the gate. The timer gives a reading of 333 m/s for the glider to pass through this gate. Friction is negligible. What is the acceleration (assumed constant) of the glider along the track?
To find the acceleration of the glider along the track, we can use the kinematic equation:
\(v^2 = u^2 + 2as\)
where:
\(v\) = final velocity of the glider (333 m/s)
\(u\) = initial velocity of the glider (0 m/s because it is released from rest)
\(s\) = displacement of the glider (96 cm or 0.96 m)
\(a\) = acceleration
Rearranging the equation, we get:
\(a = \frac{{v^2 - u^2}}{{2s}}\)
Substituting the given values, we can calculate the acceleration:
\(a = \frac{{(333\, \text{m/s})^2 - (0\, \text{m/s})^2}}{{2 \times 0.96\, \text{m}}}\)
\(a = \frac{{110889\, \text{m}^2/\text{s}^2}}{{1.92\, \text{m}}}\)
\(a \approx 57615\, \text{m/s}^2\)
Therefore, the acceleration of the glider along the track is approximately \(57615\, \text{m/s}^2\).
To find the acceleration of the glider along the track, we can use the equations of motion and the given information. Let's break down the problem step by step:
Step 1: Find the time it takes for the glider to pass through the gate.
The timer reading of 333 m/s gives us the speed, not the time. We need to find the time by dividing the distance traveled by the speed.
Distance traveled = 96 cm = 0.96 m
Speed = 333 m/s
Time = Distance / Speed
Time = 0.96 m / 333 m/s
Step 2: Calculate the velocity of the glider.
We know that velocity is the change in distance over the change in time.
Velocity = Distance / Time
Velocity = 0.96 m / (0.96 m / 333 m/s)
Step 3: Calculate the average velocity of the glider.
Since the acceleration is assumed to be constant, we can use the average velocity formula to find the acceleration.
Average Velocity = (Initial Velocity + Final Velocity) / 2
The glider starts from rest, so the initial velocity is 0 m/s.
Average Velocity = (0 m/s + Final Velocity) / 2
Step 4: Find the final velocity of the glider.
The final velocity of the glider can be calculated using the formula:
Final Velocity = Initial Velocity + (Acceleration × Time)
The glider starts from rest, so the initial velocity is 0 m/s.
Final Velocity = 0 m/s + (Acceleration × (0.96 m / (0.96 m / 333 m/s)))
Step 5: Relate the final velocity to the average velocity.
By substituting the final velocity found in step 4 into the average velocity formula from step 3, we can calculate the acceleration.
Average Velocity = (0 m/s + Final Velocity) / 2
Now we can solve for acceleration by equating the two average velocity expressions:
(0 m/s + Final Velocity) / 2 = (0 m/s + Final Velocity) / 2
Since the right side of the equation is equal to itself, we conclude that the acceleration is constant.
Therefore, the acceleration of the glider along the track is 0 m/s².