An object starts from rest at x = 0 m and moves with a constant acceleration of 3 m/s2 along the x-axis. During its journey from x = 13.5 m to x = 54 m, its average velocity is:
Understanding Average Velocity with Constant Acceleration
The question asks us to find the average velocity of an object undergoing constant acceleration during a specific part of its journey. The object starts from rest at the origin and moves along the x-axis with a constant acceleration of \(3\ m/s^2\).
Average velocity is defined as the total displacement divided by the total time taken for that displacement.
\[ v_{avg} = \frac{\text{Total Displacement}}{\text{Total Time Taken}} \]
In this problem, the object's journey is from \(x_1 = 13.5\ m\) to \(x_2 = 54\ m\). The total displacement during this part of the journey is \(\Delta x = x_2 - x_1\).
To find the total time taken, we need to determine the time the object takes to reach \(x = 13.5\ m\) and the time it takes to reach \(x = 54\ m\), starting from rest at \(x = 0\ m\).
Calculating Time at Specific Positions
Since the object starts from rest (\(v_0 = 0\)) at \(x_0 = 0\ m\) and moves with constant acceleration (\(a = 3\ m/s^2\)), we can use the kinematic equation:
\[ x = x_0 + v_0 t + \frac{1}{2} a t^2 \]
Substituting \(x_0 = 0\) and \(v_0 = 0\), the equation simplifies to:
\[ x = \frac{1}{2} a t^2 \]
Time to Reach x = 13.5 m
Let \(t_1\) be the time taken to reach \(x_1 = 13.5\ m\). Using the simplified kinematic equation:
\[ 13.5 = \frac{1}{2} (3) t_1^2 \]
\[ 13.5 = 1.5 t_1^2 \]
\[ t_1^2 = \frac{13.5}{1.5} = 9 \]
\[ t_1 = \sqrt{9} = 3\ s \]
(We take the positive root as time is positive).
Time to Reach x = 54 m
Let \(t_2\) be the time taken to reach \(x_2 = 54\ m\). Using the same kinematic equation:
\[ 54 = \frac{1}{2} (3) t_2^2 \]
\[ 54 = 1.5 t_2^2 \]
\[ t_2^2 = \frac{54}{1.5} = 36 \]
\[ t_2 = \sqrt{36} = 6\ s \]
(We take the positive root).
Calculating Displacement and Time Interval
The displacement during the journey from \(x = 13.5\ m\) to \(x = 54\ m\) is:
\[ \Delta x = x_2 - x_1 = 54\ m - 13.5\ m = 40.5\ m \]
The time taken for this specific part of the journey is the difference between the times \(t_2\) and \(t_1\):
\[ \Delta t = t_2 - t_1 = 6\ s - 3\ s = 3\ s \]
Calculating Average Velocity
Now we can calculate the average velocity using the formula:
\[ v_{avg} = \frac{\Delta x}{\Delta t} = \frac{40.5\ m}{3\ s} \]
\[ v_{avg} = 13.5\ m/s \]
Summary of the Calculation
The object starts from rest, accelerates constantly, and we calculated the time it takes to reach 13.5 m and 54 m. We found the displacement between these points and the time interval for this displacement. Using these values, we determined the average velocity for the journey from 13.5 m to 54 m.
The calculated average velocity is \(13.5\ m/s\).
Revision Table: Key Physics Concepts
| Concept | Definition | Relevant Formula (for constant acceleration) |
|---|---|---|
| Average Velocity | Total displacement divided by the total time taken. | \(v_{avg} = \frac{\Delta x}{\Delta t}\) |
| Constant Acceleration | Velocity changes by the same amount in every equal time interval. | \(a = \frac{\Delta v}{\Delta t}\) (constant) |
| Displacement | Change in position of an object. | \(\Delta x = x_f - x_i\) |
| Kinematic Equation | Relates displacement, velocity, acceleration, and time for motion with constant acceleration. | \(x = x_0 + v_0 t + \frac{1}{2} a t^2\) \(v = v_0 + at\) \(v^2 = v_0^2 + 2a(x - x_0)\) |
Additional Information on Constant Acceleration Motion
When an object moves with constant acceleration, its instantaneous velocity is constantly changing. However, the average velocity over any time interval can be calculated using the total displacement and the total time taken.
For constant acceleration motion, there are several useful kinematic equations that relate the initial velocity (\(v_0\)), final velocity (\(v\)), displacement (\(\Delta x\) or \(x-x_0\)), acceleration (\(a\)), and time (\(t\)). The equation \(x = x_0 + v_0 t + \frac{1}{2} a t^2\) was crucial in this problem because it directly relates position, initial velocity, acceleration, and time, allowing us to find the time taken to reach specific positions.
It's important to distinguish between average velocity and instantaneous velocity. Instantaneous velocity is the velocity of the object at a precise moment in time, while average velocity is calculated over a duration of time.
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