A particle moves along the x-axis so that at time t >= 0 its position is given by x(t)=-t^(3)+8t^(2)-20 t. Determine the speed of the particle at t=1. Answer:

Identify Position Function: Identify the position function and the time at which we need to find the speed. The position function is given by x(t) = -t^3 + 8t^2 - 20t, and we need to find the speed at t = 1.Recall Speed Definition: Recall that speed is the absolute value of velocity, and velocity is the derivative of the position function with respect to time. We need to find the derivative of x(t) to get the velocity function v(t).Differentiate Position Function: Differentiate the position function x(t) with respect to t to find the velocity function v(t). v(t) = dx/dt = d/dt(-t^3 + 8t^2 - 20t) = -3t^2 + 16t - 20.Evaluate Velocity at t=1: Evaluate the velocity function at t = 1 to find the velocity at that time. v(1) = -3(1)^2 + 16(1) - 20 = -3 + 16 - 20 = -7.Find Speed at t=1: Find the speed by taking the absolute value of the velocity at t = 1. Speed at t = 1 is |v(1)| = |-7| = 7.

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A particle moves along the
x-axis so that at time
t >= 0 its position is given by
x(t)=-t^(3)+8t^(2)-20 t. Determine the speed of the particle at
t=1.
Answer:

A particle moves along the $x$ -axis so that at time $t \geq 0$ its position is given by $x(t)=-t^{3}+8 t^{2}-20 t$ . Determine the speed of the particle at $t=1$ . $\newline$ Answer:

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Math Problems

Calculus

Relate position, velocity, speed, and acceleration using derivatives

Full solution

Q. A particle moves along the

x

-axis so that at time

t \geq 0

its position is given by

x(t)=-t^{3}+8 t^{2}-20 t

. Determine the speed of the particle at

t=1

\newline

Answer:

Identify Position Function: Identify the position function and the time at which we need to find the speed. The position function is given by $x(t) = -t^3 + 8t^2 - 20t$ , and we need to find the speed at $t = 1$ .
Recall Speed Definition: Recall that speed is the absolute value of velocity, and velocity is the derivative of the position function with respect to time. We need to find the derivative of $x(t)$ to get the velocity function $v(t)$ .
Differentiate Position Function: Differentiate the position function $x(t)$ with respect to $t$ to find the velocity function $v(t)$ . $v(t) = \frac{dx}{dt} = \frac{d}{dt}(-t^3 + 8t^2 - 20t) = -3t^2 + 16t - 20$ .
Evaluate Velocity at $t=1$ : Evaluate the velocity function at $t = 1$ to find the velocity at that time. $v(1) = -3(1)^2 + 16(1) - 20 = -3 + 16 - 20 = -7.$
Find Speed at $t=1$ : Find the speed by taking the absolute value of the velocity at $t = 1$ . Speed at $t = 1$ is $|v(1)| = |-7| = 7$ .