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The equation of motion for a float in a parade is \(x=-9.2 \mathrm{~m}+(1.5 \mathrm{~m} / \mathrm{s}) t\). (a) Where is the float at \(t=3.5 \mathrm{~s}\) ? (b) At what time is the float at \(x=0\) ?

Short Answer

Expert verified
At \(t = 3.5\) s, the float is at \(-3.95\) m; it reaches\(x = 0\) m at \(t \approx 6.13\) s.

Step by step solution

01

Identify the given equation and interpret

The equation given is the equation of motion for the float: \[x = -9.2 \, \text{m} + (1.5 \, \text{m/s}) t\]Here, "\( x \)" represents the position of the float as a function of time "\( t \)", measured in meters and seconds, respectively. We will use this equation to find the float's position at a specific time and the time it reaches a position of zero meters.
02

Calculate float's position at specified time

To find where the float is at \(t = 3.5 \, \mathrm{s}\), substitute \(t = 3.5\) into the equation:\[x = -9.2 + (1.5 \cdot 3.5)\]Calculate:\[x = -9.2 + 5.25 = -3.95 \, \mathrm{m}\]Thus, at \(t = 3.5 \, \mathrm{s}\), the float is at \(-3.95 \, \mathrm{m}\).
03

Determine the time when float is at x = 0

To find when the float is at \(x = 0\), set the equation to zero and solve for \(t\):\[0 = -9.2 + 1.5t\]Add 9.2 to both sides:\[9.2 = 1.5t\]Divide both sides by 1.5:\[t = \frac{9.2}{1.5} \approx 6.13 \, \mathrm{s}\]Thus, the float reaches \(x = 0\) at approximately \(t = 6.13 \, \mathrm{s}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation of Motion
The equation of motion is a fundamental tool in kinematics. It helps describe how an object's position changes over time. In our exercise, the equation for the float is:
\[x = -9.2 \text{ m} + (1.5 \text{ m/s}) t\]
This simple linear equation tells us that the float starts at an initial position of \(-9.2\) meters.
The presence of the "\(+ (1.5 \text{ m/s}) t\)" part indicates a uniform motion, with a constant speed of \(1.5\) meters per second in the positive direction.
Using these types of equations allows us to predict an object's future or past positions based on given conditions.
  • Initial Position: The value where the object starts, often represented by a constant term, like \(-9.2\) in this case.
  • Rate of Change: Represented by the coefficient of \(t\), indicating the speed at which position changes over time.
Time Calculation
Calculating time in kinematics involves determining when an object reaches a certain position. From our exercise, the expression \(x= 0\) means we are solving for the time when the float is exactly at the start line or reference point.
By setting the motion equation to zero, we reverse-engineer the formula to isolate \(t\):
\[0 = -9.2 + 1.5t\]
Adding 9.2 to both sides gives us the distance traveled due to motion, which yields:
\[9.2 = 1.5t\]
Finally, dividing both sides by 1.5 will solve for time:
  • Calculate: \[ t = \frac{9.2}{1.5} \approx 6.13 \, \text{s}\]
This means the float reaches the starting position or point \(x=0\) at \(6.13\) seconds. Understanding this process involves recognizing the relationship between position, time, and constant speed in linear motion.
Displacement Calculation
Displacement refers to an object's change in position. As used in the exercise, displacement calculation means finding out where the float is at a specific time. For instance, determining the float's position at \(t=3.5\) seconds involves substituting the time in the equation of motion.
For the float, the formula is substituted as follows:
  • Substitute time in equation: \[x = -9.2 + (1.5 \times 3.5)\]
  • Simplify the calculation: \[x = -9.2 + 5.25 = -3.95 \, \text{m}\]
This tells us that at \(3.5\) seconds, the float is at \(-3.95\) meters.
This information not only gives the float's current position but also the relative position to its start, effectively showcasing the float's displacement.

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Most popular questions from this chapter

Predict \& Explain You drive your car in a straight line at \(15 \mathrm{~m} / \mathrm{s}\) for \(10 \mathrm{~km}\), then at \(25 \mathrm{~m} / \mathrm{s}\) for another \(10 \mathrm{~km}\). (a) Is your average speed for the entire trip more than, less than, or equal to \(20 \mathrm{~m} / \mathrm{s}\) ? (b) Choose the best explanation from the following: A. More time is spent driving at \(15 \mathrm{~m} / \mathrm{s}\) than at \(25 \mathrm{~m} / \mathrm{s}\). B. The average of \(15 \mathrm{~m} / \mathrm{s}\) and \(25 \mathrm{~m} / \mathrm{s}\) is \(20 \mathrm{~m} / \mathrm{s}\). C. Less time is spent driving at \(15 \mathrm{~m} / \mathrm{s}\) than at \(25 \mathrm{~m} / \mathrm{s}\).

Triple Choice An object's position-time graph is a straight line with a negative slope. Is the speed of this object positive, negative, or zero? Explain.

The position-time equation of motion for a bunny hopping across a yard is $$ x_{\mathrm{f}}=8.3 \mathrm{~m}+(2.2 \mathrm{~m} / \mathrm{s}) t $$ (a) What is the initial position of the bunny? (b) What is the bunny's velocity? 36\. A bowling ball moves with constant velocity from an initial position of \(1.6 \mathrm{~m}\) to a final position of \(7.8 \mathrm{~m}\) in \(3.1 \mathrm{~s}\). (a) What is the position-time equation for the bowling ball? (b) At what time is the ball at the position \(8.6 \mathrm{~m}\) ?

Think \& Calculate A train travels in a straight line at \(20.0 \mathrm{~m} / \mathrm{s}\) for \(2 \mathrm{~km}\), then at \(30.0 \mathrm{~m} / \mathrm{s}\) for another \(2 \mathrm{~km}\). (a) Is the average speed of the train greater than, less than, or equal to \(25 \mathrm{~m} / \mathrm{s}\) ? Explain. (b) Verify your answer to part (a) by calculating the average speed.

A golf cart moves with a velocity of \(8 \mathrm{~m} / \mathrm{s}\). Is the displacement of the golf cart from \(t=0\) to \(t=5 \mathrm{~s}\) greater than, less than, or equal to its displacement from \(t=5 \mathrm{~s}\) to \(t=10 \mathrm{~s}\) ? Explain.

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