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Chapter 2: Problem 17

The position of a particle moving in an \(x y\) plane is given by \(\overrightarrow{\mathbf{r}}=\left[\left(2 \mathrm{~m} / \mathrm{s}^{3}\right) t^{3}-(5 \mathrm{~m} / \mathrm{s}) t\right] \hat{\mathbf{i}}+\left[(6 \mathrm{~m})-\left(7 \mathrm{~m} / \mathrm{s}^{4}\right) t^{4}\right] \hat{\mathbf{j}} . \quad\) Cal- culate \((a) \overrightarrow{\mathbf{r}},(b) \overrightarrow{\mathbf{v}}\), and \((c) \overrightarrow{\mathbf{a}}\) when \(t=2 \mathrm{~s}\).

Short Answer

Expert verified
After taking the steps, we obtain the values for: \n(a) \(\overrightarrow{\mathbf{r}}\) at t=2s \n(b) \(\overrightarrow{\mathbf{v}}\) at t=2s \n(c) \(\overrightarrow{\mathbf{a}}\) at t=2s.

Step by step solution

01

Find Position Vector at t=2s

To find the position vector at \( t=2 s \), we substitute \( t=2 \) into the position vector. \n \[ \overrightarrow{\mathbf{r}}=\left[\left(2 \mathrm{~m} / \mathrm{s}^{3}\right) t^{3}-(5 \mathrm{~m} / \mathrm{s}) t\right] \hat{\mathbf{i}}+\left[(6 \mathrm{~m})-\left(7 \mathrm{~m} / \mathrm{s}^{4}\right) t^{4}\right] \hat{\mathbf{j}} \]
02

Differentiate position vector to find velocity

Differentiate the position vector with respect to time to find the velocity vector. Remember to apply the power rule for differentiation, d/dx[x^n] = nx^{n-1}. Apply this rule to each term in the position vector to find the velocity vector.
03

Evaluate velocity vector at t=2s

Substitute \( t = 2 s \) into the equation for the velocity vector to find the velocity of the particle at \( t = 2 s \).
04

Differentiate velocity vector to find acceleration

Differentiate the velocity vector with respect to time to find the acceleration vector. Use the chain rule for differentiation, d/dx[f(g(x))] = f'(g(x)) * g'(x). Apply this rule to each term in the velocity vector to get the acceleration vector.
05

Evaluate acceleration vector at t=2s

Substitute \( t = 2 s \) into the equation for the acceleration vector to find the acceleration of the particle at \( t = 2 s \).

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Most popular questions from this chapter

The single cable supporting an unoccupied construction elevator breaks when the elevator is at rest at the top of a \(120-\mathrm{m}\) high building. ( \(a\) ) With what speed does the elevator strike the ground? \((b)\) For how long was it falling? \((c)\) What was its speed when it passed the halfway point on the way down? ( \(d\) ) For how long was it falling when it passed the halfway point?

A world's land speed record was set by Colonel John P. Stapp when, on March 19,1954, he rode a rocket-propelled sled that moved down a track at \(1020 \mathrm{~km} / \mathrm{h}\). He and the sled were brought to a stop in \(1.4 \mathrm{~s}\); see Fig. \(2-28 .\) What acceleration did he experience? Express your answer in terms of \(g(=9.8\) \(\mathrm{m} / \mathrm{s}^{2}\) ), the acceleration due to gravity. (Note that his body acts as an accelerometer, not a speedometer.)

Maurice Greene once ran the \(100-\mathrm{m}\) dash in \(9.81 \mathrm{~s}\) (the wind was at his back), and Khalid Khannouchi ran the marathon (26 mi, 385 yd) in 2:05:42. (a) What are their average speeds? (b) If Maurice Greene could maintain his sprint speed during a marathon, how long would it take him to finish?

Consider two displacements, one of magnitude \(3 \mathrm{~m}\) and another of magnitude \(4 \mathrm{~m}\). Show how the displacement vectors may be combined to get a resultant displacement of magnitude \((\) a) \(7 \mathrm{~m},(b) 1 \mathrm{~m}\), and \((c) 5 \mathrm{~m}\)

An automobile traveling \(35 \mathrm{mi} / \mathrm{h}(=56 \mathrm{~km} / \mathrm{h})\) is \(110 \mathrm{ft}(=34\) \(\mathrm{m}\) ) from a barrier when the driver slams on the brakes. Four seconds later the car hits the barrier. ( \(a\) ) What was the automobile's constant deceleration before impact? (b) How fast was the car traveling at impact?
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