Problem 5 - Fundamentals of Physics

Question

The position of an object moving along an \(x\) axis is given by \(x=3 t-4 t^{2}+t^{3},\) where \(x\) is in meters and \(t\) in seconds. Find the position of the object at the following values of \(t:\) (a) \(1 \mathrm{~s}\). (b) \(2 \mathrm{~s}\). (c) \(3 \mathrm{~s}\), and (d) \(4 \mathrm{~s}\). (c) What is the object's dicplacement hetween \(t=0\) and \(t=4 \mathrm{~s} ?\) (f) What is its average velocity for the time interval from \(t=2 \mathrm{~s}\) to \(t=4 \mathrm{~s} ?\) (g) Graph \(x\) versus \(t\) for \(0 \leq t \leq 4 \mathrm{~s}\) and indicate how the answer for (f) can be found on the graph.

Step-by-Step Solution

Answer

(a) 0 m, (b) -2 m, (c) 0 m, (d) 12 m. Displacement: 12 m. Average velocity: 7 m/s.

Step 1: Calculate Position at t=1s

Substitute \( t = 1 \) into the position function \( x(t) = 3t - 4t^2 + t^3 \). This gives us \( x(1) = 3(1) - 4(1)^2 + (1)^3 = 3 - 4 + 1 = 0 \). Therefore, the position at \( t = 1 \text{ s} \) is \( x = 0 \) meters.

Step 2: Calculate Position at t=2s

Substitute \( t = 2 \) into \( x(t) = 3t - 4t^2 + t^3 \). This yields \( x(2) = 3(2) - 4(2)^2 + (2)^3 = 6 - 16 + 8 = -2 \). The position at \( t = 2 \text{ s} \) is \( x = -2 \) meters.

Step 3: Calculate Position at t=3s

Substitute \( t = 3 \) into the function: \( x(3) = 3(3) - 4(3)^2 + (3)^3 = 9 - 36 + 27 = 0 \). Thus, the position at \( t = 3 \text{ s} \) is \( x = 0 \) meters.

Step 4: Calculate Position at t=4s

Calculate \( x(4) \) by substituting \( t = 4 \) into the equation: \( x(4) = 3(4) - 4(4)^2 + (4)^3 = 12 - 64 + 64 = 12 \). Therefore, the position at \( t = 4 \text{ s} \) is \( x = 12 \) meters.

Step 5: Calculate Displacement from t=0s to t=4s

The displacement is the change in position from \( t = 0 \text{ s} \) to \( t = 4 \text{ s} \). First calculate \( x(0) = 3(0) - 4(0)^2 + (0)^3 = 0 \). The displacement \( \Delta x = x(4) - x(0) = 12 - 0 = 12 \) meters.

Step 6: Calculate Average Velocity from t=2s to t=4s

The average velocity is given by \( v_{avg} = \frac{x(4) - x(2)}{4 - 2} = \frac{12 - (-2)}{2} = \frac{14}{2} = 7 \text{ m/s} \). So the average velocity between \( t = 2 \) s and \( t = 4 \) s is \( 7 \text{ m/s} \).

Step 7: Graph x vs t

To graph \( x(t) \), plot the points \((0, 0), (1, 0), (2, -2), (3, 0), (4, 12)\). Draw a smooth curve through these points. The average velocity from \( t = 2 \text{ s} \) to \( t = 4 \text{ s} \) is the slope of the line joining the points \((2, -2)\) and \((4, 12)\) on the graph.

Key Concepts

Displacement

Understanding displacement is key to analyzing motion. Displacement refers to the change in position of an object. It's a vector quantity, which means it has both magnitude and direction. Displacement is different from distance, as it only considers the shortest path between two points, rather than the entire path traveled.
To calculate displacement, you need to find the change in the position function:

• Identify the initial position using the position function at the start time, which is \(t = 0\).
• Find the final position at the end time, here \(t = 4 \, \text{s}\).
• Subtract the initial position from the final position to find displacement: \(\Delta x = x(4) - x(0)\).

For example, as calculated previously, the object's displacement between \(t = 0\) and \(t = 4 \, \text{s}\) is \(12\, \text{meters}\). This means the object's position changed by 12 meters in the positive direction over this time interval.

Average Velocity

Average velocity gives a measure of how fast an object is moving over a particular time interval. Like displacement, average velocity is a vector quantity, which means it reflects both speed and direction. To find average velocity, follow these steps:

• Subtract the initial position from the final position to find the displacement over the time interval.
• Divide this displacement by the time interval length.

Mathematically, it is given by the formula: \(v_{\text{avg}} = \frac{x(\text{final}) - x(\text{initial})}{t_{\text{final}} - t_{\text{initial}}}\).
For example, between the times \(t = 2\, \text{s}\) and \(t = 4\, \text{s}\), the average velocity was calculated as \(7\, \text{m/s}\). This tells us that on average, the object moved 7 meters every second in this interval.

Graphing Motion

Graphing motion gives a visual representation of how an object's position changes over time, making concepts easier to understand. When you graph position versus time, you're typically plotting points of the position function's outputs (the y-axis) against time inputs (the x-axis).
Here's how you can graph motion:

• Plot the calculated positions at specified time intervals, as done by \(t = 0, 1, 2, 3, 4\).
• Connect these points to form a smooth curve or line, reflecting the object's motion.

By looking at the graph, you can also determine the average velocity by evaluating the slope of the line between two points within the time interval. For instance, between \(t = 2\, \text{s}\) and \(t = 4\, \text{s}\), the line or curve segment should visually represent the 7 m/s average velocity calculated earlier. This visual aid helps in understanding how quickly or slowly the object's position is changing over time.