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Chapter 12: Problem 31

The velocity of a particle traveling along a straight line is \(v=v_{0}-k s\), where \(k\) is constant. If \(s=0\) when \(t=0\), determine the position and acceleration of the particle as a function of time.

Short Answer

Expert verified
The position of the particle at time \(t\) is given by \(s = v_{0}/k - e^{(-kt)}\) and the acceleration at time \(t\) is given by \(a(t) =ke^{(-kt)}\).

Step by step solution

01

Solving for Position

Re-arrange the equation to get \(ds = dt( v_{0}-k s)\), then integrate both sides. The integral of the left side will give the position \(s\) and the right side will give the expression for time \(t\), leading to: \(s = v_{0}/k - e^{(-kt)}\).
02

Solving for Acceleration

The acceleration is given by \(a(t) = dv/dt = -k ds/dt\). By substituting the equation for position obtained from step 1 into this equation and differentiating we get: \(a(t) =ke^{(-kt)}\).

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Most popular questions from this chapter

The roller coaster car travels down the helical path at constant speed such that the parametric equations that define its position are \(x=c \sin k t, y=c \cos k t, z=h-b t\), where \(c, h\), and \(b\) are constants. Determine the magnitudes of its velocity and acceleration.

The car travels along the circular path such that its speed is increased by \(a_{t}=\left(0.5 e^{t}\right) \mathrm{m} / \mathrm{s}^{2}\), where \(t\) is in seconds. Determine the magnitudes of its velocity and acceleration after the car has traveled \(s=18 \mathrm{~m}\) starting from rest. Neglect the size of the car.

At the instant shown, cars \(A\) and \(B\) are traveling at velocities of \(40 \mathrm{~m} / \mathrm{s}\) and \(30 \mathrm{~m} / \mathrm{s}\), respectively. If \(A\) is increasing its speed at \(4 \mathrm{~m} / \mathrm{s}^{2}\), whereas the speed of \(B\) is decreasing at \(3 \mathrm{~m} / \mathrm{s}^{2}\), determine the velocity and acceleration of \(B\) with respect to \(A\). The radius of curvature at \(B\) is \(\rho_{B}=200 \mathrm{~m}\).

The automobile is originally at rest at \(s=0 .\) If its speed is increased by \(\dot{v}=\left(0.05 t^{2}\right) \mathrm{ft} / \mathrm{s}^{2}\), where \(t\) is in seconds, determine the magnitudes of its velocity and acceleration when \(t=18 \mathrm{~s}\).

An airplane is flying in a straight line with a velocity of \(200 \mathrm{mi} / \mathrm{h}\) and an acceleration of \(3 \mathrm{mi} / \mathrm{h}^{2}\). If the propeller has a diameter of \(6 \mathrm{ft}\) and is rotating at a constant angular rate of \(120 \mathrm{rad} / \mathrm{s}\), determine the magnitudes of velocity and acceleration of a particle located on the tip of the propeller.
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