A particle travels along the $x$ -axis such that its acceleration is given by $a(t)=t^{0.5} \sin (2 t)$ . If the velocity of the particle is $v=-3$ when $t=1$ , what is the velocity of the particle when $t=4$ ? You may use a calculator and round your answer to the nearest thousandth. $\newline$ Answer:

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Relate position, velocity, speed, and acceleration using derivatives

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Q. A particle travels along the

x

-axis such that its acceleration is given by

a(t)=t^{0.5} \sin (2 t)

. If the velocity of the particle is

v=-3

when

t=1

, what is the velocity of the particle when

t=4

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Answer:

Write Integral Expression: To find the velocity at $t=4$ , we need to integrate the acceleration function from $t=1$ to $t=4$ . The velocity function $v(t)$ is the integral of the acceleration function $a(t)$ .
Evaluate Integral: First, we write down the integral that we need to evaluate: $\newline$ $v(t) = \int_{t=1}^{t=4} a(t) \, dt$ $\newline$ $v(t) = \int_{t=1}^{t=4} t^{0.5}\sin(2t) \, dt$
Calculate Final Velocity: We can use a calculator to evaluate the definite integral. The result of the integral will give us the change in velocity from $t=1$ to $t=4$ .
Calculate Final Velocity: We can use a calculator to evaluate the definite integral. The result of the integral will give us the change in velocity from $t=1$ to $t=4$ .After calculating the integral, we add the initial velocity at $t=1$ to the result of the integral to find the velocity at $t=4$ . $v(4) = v(1) + \int_{1}^{4} t^{(0.5)}\sin(2t) \, dt$ $v(4) = -3 + [\text{result of the integral}]$
Calculate Final Velocity: We can use a calculator to evaluate the definite integral. The result of the integral will give us the change in velocity from $t=1$ to $t=4$ .After calculating the integral, we add the initial velocity at $t=1$ to the result of the integral to find the velocity at $t=4$ . $v(4) = v(1) + \int_{1}^{4} t^{0.5}\sin(2t) \, dt$ $v(4) = -3 + [\text{result of the integral}]$ Using a calculator, we find the result of the integral $\int_{1}^{4} t^{0.5}\sin(2t) \, dt$ to be approximately $4.477$ (rounded to the nearest thousandth).
Calculate Final Velocity: We can use a calculator to evaluate the definite integral. The result of the integral will give us the change in velocity from $t=1$ to $t=4$ .After calculating the integral, we add the initial velocity at $t=1$ to the result of the integral to find the velocity at $t=4$ . $v(4) = v(1) + \int_{1}^{4} t^{0.5}\sin(2t) \, dt$ $v(4) = -3 + [\text{result of the integral}]$ Using a calculator, we find the result of the integral $\int_{1}^{4} t^{0.5}\sin(2t) \, dt$ to be approximately $4.477$ (rounded to the nearest thousandth).Now we add the initial velocity to the result of the integral to find the final velocity at $t=4$ . $v(4) = -3 + 4.477$ $v(4) = 1.477$