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Chapter 2: Problem 12

A sprinter explodes out of the starting block with an acceleration of \(+2.3 \mathrm{~m} / \mathrm{s}^{2},\) which she sustains for \(1.2 \mathrm{~s}\). Then, her acceleration drops to zero for the rest of the race. What is her velocity (a) at \(t=1.2 \mathrm{~s}\) and \((\mathrm{b})\) at the end of the race?

Short Answer

Expert verified
(a) 2.76 m/s, (b) 2.76 m/s

Step by step solution

01

Identify Given Information

We have an initial acceleration of the sprinter being \(a_1 = +2.3 \ \mathrm{m/s}^2\) for the first \(1.2\) seconds. After this time, her acceleration becomes \(a_2 = 0\ \mathrm{m/s}^2\) for the rest of the race. Initially, her velocity is \(v_0 = 0\ \mathrm{m/s}\) at \(t=0\).
02

Calculate Velocity at t = 1.2 s

We will use the formula for velocity when an object moves with uniform acceleration, \(v = v_0 + at\). Applying the values, \(v = 0 \ \mathrm{m/s} + (2.3 \ \mathrm{m/s}^2)(1.2 \ \mathrm{s}) = 2.76 \ \mathrm{m/s}\). Therefore, the velocity at \(t=1.2\) seconds is \(2.76 \ \mathrm{m/s}\).
03

Determine the Velocity at the End of the Race

Since her acceleration is zero after \(t=1.2\) seconds, her velocity remains constant for the rest of the race. Thus, the velocity at the end of the race is the same as the velocity just after \(1.2\) seconds, which is \(2.76 \ \mathrm{m/s}\).

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Key Concepts

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

Acceleration
Acceleration is a key concept in kinematics. It refers to how quickly an object's velocity changes over time. Imagine you're driving a car, and you push on the gas pedal. As the car speeds up, it's experiencing acceleration. In the context of the sprinter, her acceleration is initially given as \(+2.3 \, \text{m/s}^2\), which means her speed increases by \(2.3\) meters per second every second for a brief period of \(1.2\) seconds.

Understanding acceleration helps us describe how the sprinter's motion changes as she moves down the track. During this initial phase, she gets faster and faster. But after the \(1.2\) seconds, her acceleration drops to zero. This means she stops increasing in speed and thus maintains whatever speed she reached by then.
  • Positive acceleration means the speed is increasing.
  • Negative acceleration (often called deceleration) means slowing down.
  • Zero acceleration indicates a constant speed.
Velocity
Velocity is another fundamental concept of motion and a key part of kinematics. Simply put, it describes the speed of an object in a given direction. If you imagine the sprinter again, her velocity is initially \(0 \, \text{m/s}\) because she starts from rest.

In the first \(1.2\) seconds, due to her acceleration, the sprinter's velocity changes. This can be calculated using the formula for velocity with uniform acceleration, \(v = v_0 + at\). By plugging in the known values, we find her velocity at \(t = 1.2\) seconds to be \(2.76 \, \text{m/s}\).
  • Velocity has both a magnitude (speed) and a direction.
  • It is a vector quantity.
  • When acceleration ceases after \(1.2\) seconds, her velocity stays constant.
Uniform Motion
Uniform motion occurs when an object moves at a constant speed in a straight line. It's the simplest form of motion because the velocity doesn't change. In this exercise, after the initial \(1.2\) seconds of acceleration, the sprinter transitions into uniform motion.

This means her speed remains at \(2.76 \, \text{m/s}\). There is no acceleration impacting her anymore, and thus, she continues moving at that constant velocity for the remainder of the race.
  • Uniform motion implies zero acceleration.
  • The object covers equal distances in equal intervals of time.
  • It's characterized by a straight line on a velocity vs. time graph.

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Most popular questions from this chapter

(a) What is the magnitude of the average acceleration of a skier who, starting from rest, reaches a speed of \(8.0 \mathrm{~m} / \mathrm{s}\) when going down a slope for \(5.0 \mathrm{~s} ?\) (b) How far does the skier travel in this time?

A tourist being chased by an angry bear is running in a straight line toward his car at a speed of \(4.0 \mathrm{~m} / \mathrm{s}\). The car is a distance \(d\) away. The bear is \(26 \mathrm{~m}\) behind the tourist and running at \(6.0 \mathrm{~m} / \mathrm{s}\). The tourist reaches the car safely. What is the maximum possible value for \(d ?\)

A speedboat starts from rest and accelerates at \(+2.01 \mathrm{~m} / \mathrm{s}^{2}\) for \(7.00 \mathrm{~s}\). At the end of this time, the boat continues for an additional \(6.00 \mathrm{~s}\) with an acceleration of \(+0.518 \mathrm{~m} / \mathrm{s}^{2}\) Following this, the boat accelerates at \(-1.49 \mathrm{~m} / \mathrm{s}^{2}\) for \(8.00 \mathrm{~s}\). (a) What is the velocity of the boat at \(t=21.0 \mathrm{~s} ?\) (b) Find the total displacement of the boat.

In getting ready to slam-dunk the ball, a basketball player starts from rest and sprints to a speed of \(6.0 \mathrm{~m} / \mathrm{s}\) in \(1.5 \mathrm{~s}\). Assuming that the player accelerates uniformly, determine the distance he runs.

The initial velocity \(v_{0}\) and acceleration \(a\) of four moving objects at a given instant in time are given in the table: $$ \begin{array}{|c|c|c|} \hline & \text { Initial velocity } v_{0} & \text { Acceleration } a \\ \hline \text { (a) } & +12 \mathrm{~m} / \mathrm{s} & +3.0 \mathrm{~m} / \mathrm{s}^{2} \\ \hline \text { (b) } & +12 \mathrm{~m} / \mathrm{s} & -3.0 \mathrm{~m} / \mathrm{s}^{2} \\ \hline \text { (c) } & -12 \mathrm{~m} / \mathrm{s} & +3.0 \mathrm{~m} / \mathrm{s}^{2} \\ \hline \text { (d) } & -12 \mathrm{~m} / \mathrm{s} & -3.0 \mathrm{~m} / \mathrm{s}^{2} \\ \hline \end{array} $$ Draw vectors for \(v_{0}\) and \(a\) and, in each case, state whether the speed of the object is increasing or decreasing in time. Account for your answers. Problem For each of the four pairs in the table above, determine the final speed of the object if the elapsed time is \(2.0 \mathrm{~s}\). Compare your final speeds with the initial speeds and make sure that your answers are consistent with your answers to the Concept Questions.
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