Chapter 2: Q. 40 (page 61)

A particle’s velocity is described by the function vx = kt 2 m/s, where k is a constant and t is in s. The particle’s position at t0 = 0 s is x0 = - 9.0 m. At t1 = 3.0 s, the particle is at x1 = 9.0 m. Determine the value of the constant k. Be sure to include the proper units

Short Answer

Expert verified

The value of constant $k$ is 2.

Step by step solution

Given information

The velocity of a particle is given by a function $v_{x} = k t^{2} \frac{m}{s}$ .

The position of the particle at $t_{0} = 0 s$ is $x_{0} = - 9.0 m$ .

The position of the particle at $t_{1} = 3.0 s$ is $x_{1} = 9.0 m$ .

The value of constant k.

The position of a particle is given by

$x_{f} = x_{0} + \int_{t_{0}}^{t_{f}} v_{x} d t$

here,

role="math" localid="1648490216091" $x_{1} = x_{0} + \int_{t_{0}}^{t_{1}} v_{x} d t = x_{0} + \int_{t_{0}}^{t_{1}} k t^{2} d t = x_{0} + {[\frac{1}{3} k t^{3}]}_{t_{0}}^{t_{1}} = x_{0} + \frac{1}{3} k t_{1}^{3} - \frac{1}{3} k t_{0}^{3}$

Substitute known values

$9 = - 9 + \frac{1}{3} k {(3)}^{3} - \frac{1}{3} k {(0)}^{3} 9 = - 9 + 9 k 9 k = 18 k = 2$

Therefore, the value of constant $k$ is 2.

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A particle’s velocity is described by the function vx = kt 2 m/s, where k is a constant and t is in s. The particle’s position at t0 = 0 s is x0 = - 9.0 m. At t1 = 3.0 s, the particle is at x1 = 9.0 m. Determine the value of the constant k. Be sure to include the proper units

Short Answer

Step by step solution

Given information

The value of constant k.

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