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(a) What is the velocity of an object that moves from \(73 \mathrm{~m}\) to \(62 \mathrm{~m}\) in \(12 \mathrm{~s}\) ? (b) What is its speed?

Short Answer

Expert verified
The velocity is \(-0.9167 \,\text{m/s}\) and the speed is \(0.9167 \,\text{m/s}\).

Step by step solution

01

Identify Initial and Final Positions

First, identify the initial position \( x_i \) and the final position \( x_f \) of the object. Here, \( x_i = 73 \,\text{m} \) and \( x_f = 62 \,\text{m} \).
02

Calculate Displacement

Calculate the displacement \( \Delta x \) of the object using the formula \( \Delta x = x_f - x_i \). Substitute the values to get \( \Delta x = 62 \,\text{m} - 73 \,\text{m} = -11 \,\text{m} \).
03

Determine Time Interval

Identify the time interval \( \Delta t \) during which the displacement occurred. In this case, \( \Delta t = 12 \,\text{s} \).
04

Calculate Velocity

The velocity \( v \) is given by the displacement divided by the time interval: \( v = \frac{\Delta x}{\Delta t} \). Substitute \( \Delta x = -11 \,\text{m} \) and \( \Delta t = 12 \,\text{s} \) to calculate \( v = \frac{-11 \,\text{m}}{12 \,\text{s}} = -0.9167 \,\text{m/s} \).
05

Calculate Speed

Speed is the magnitude of velocity. Therefore, speed is the absolute value of velocity. For this problem, the speed is \( |v| = |-0.9167 \,\text{m/s}| = 0.9167 \,\text{m/s} \).

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Key Concepts

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

Displacement
When we talk about displacement, we're referring to the change in position of an object. It's how far out of place an object is; it's the object's overall change in position. Displacement can be a bit tricky because it's a vector quantity. This means it has both magnitude and direction. Knowing this helps us understand why displacement can be negative.

In our exercise, the object moves from an initial position of 73 meters to a final position of 62 meters. To find the displacement, we use the formula:
  • \( \Delta x = x_f - x_i \)
Plugging in the values, we have:
  • \( \Delta x = 62 \, \text{m} - 73 \, \text{m} = -11 \, \text{m} \)
The displacement is -11 meters because the object moved towards a lower position, or simply put, it moved backwards by 11 meters.
Time Interval
The time interval is a key concept when we want to measure how long an event takes place. It's a measure of the duration from the start to the finish of the event. This is straightforward, as it's a scalar quantity, meaning it only has magnitude and no direction.

For our exercise, the displacement occurs over a period of 12 seconds. Thinking of time intervals as the lengths of time parts of an action take can make it easier to grasp. Here, we record:
  • \( \Delta t = 12 \, \text{s} \)
By knowing this, we are equipped to calculate other important values like velocity by pairing it with displacement.
Speed Calculation
Speed is all about how fast something is moving, without considering the direction. Like time intervals, speed is a scalar quantity. When calculating speed, we look at the absolute value of velocity, which gives us only how fast the object is moving regardless of direction.

For calculating the velocity, which is the rate of change of displacement, we use the equation:
  • \( v = \frac{\Delta x}{\Delta t} \)
Substituting in our values:
  • \( v = \frac{-11 \, \text{m}}{12 \, \text{s}} = -0.9167 \, \text{m/s} \)
The negative sign indicates the direction of the velocity, backwards in this case. But for speed, we take the absolute value:
  • Speed = \(|-0.9167 \, \text{m/s}| = 0.9167 \, \text{m/s}\)
So, speed tells us how fast the object was moving, irrespective of which direction it was traveling.

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Most popular questions from this chapter

In heavy rush-hour traffic you drive in a straight line at \(12 \mathrm{~m} / \mathrm{s}\) for \(1.5 \mathrm{~min}\), then you have to stop for \(3.5 \mathrm{~min}\), and finally you drive at \(15 \mathrm{~m} / \mathrm{s}\) for another \(2.5 \mathrm{~min}\). (a) Plot a position-time graph for this motion. Your graph should extend from \(t=0\) to \(t=7.5 \mathrm{~min}\). (b) Use your graph from part (a) to calculate the average velocity between \(t=0\) and \(t=7.5 \mathrm{~min}\).

Rubber Ducks A severe storm on January 10, 1992, near the Aleutian Islands, caused a cargo ship to spill 29,000 rubber ducks and other bath toys into the ocean. Ten months later hundreds of rubber ducks began to appear along the shoreline near Sitka, Alaska, roughly \(2600 \mathrm{~km}\) away. What was the approximate average speed, in meters per second, of the ocean current that carried the ducks to shore? (Rubber ducks from the same spill began to appear on the coast of Maine in July 2003.)

Triple Choice The position-time graph for the motion of a certain particle is a smooth curve, like a parabola. At a given instant of time, the tangent line to the positiontime graph has a negative slope. Is the instantaneous velocity of the particle at this time positive, negative, or zero? Explain.

Britta Steffen of Germany set the women's Olympic record for the \(100-\mathrm{m}\) freestyle swim with a time of \(53.12 \mathrm{~s}\). What was her average speed? Give your answer in meters per second and miles per hour.

Is it possible for two different objects to have the same velocity but different initial positions?

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