A particle moves along the x-axis so that at time t >= 0 its position is given by x(t)=t^(2)+4t+15. Determine the speed of the particle at t=4. 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^2 + 4t + 15, and we need to find the speed at t = 4.Understand Speed Definition: Understand that the speed of the particle is the absolute value of the velocity, which is the derivative of the position function with respect to time. To find the speed, we first need to find the derivative of x(t) with respect to t, which will give us the velocity v(t).Calculate Derivative of Position Function: Calculate the derivative of the position function x(t) = t^2 + 4t + 15. The derivative of t^2 is 2t, the derivative of 4t is 4, and the derivative of a constant like 15 is 0. So, v(t) = dx/dt = 2t + 4.Evaluate Velocity at t = 4: Evaluate the velocity at t = 4. Substitute t = 4 into the velocity function v(t) = 2t + 4. v(4) = 2(4) + 4 = 8 + 4 = 12.Determine Speed at t = 4: Since speed is the absolute value of velocity, and the velocity at t = 4 is 12, the speed is also 12, because 12 is already a positive number. Speed at t = 4 is |v(4)| = |12| = 12.

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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^(2)+4t+15. Determine the speed of the particle at
t=4.
Answer:

A particle moves along the $x$ -axis so that at time $t \geq 0$ its position is given by $x(t)=t^{2}+4 t+15$ . Determine the speed of the particle at $t=4$ . $\newline$ Answer:

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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^{2}+4 t+15

. Determine the speed of the particle at

t=4

\newline

Answer:

Identify Position Function: Identify the position function and the time at which we need to find the speed. $\newline$ The position function is given by $x(t) = t^2 + 4t + 15$ , and we need to find the speed at $t = 4$ .
Understand Speed Definition: Understand that the speed of the particle is the absolute value of the velocity, which is the derivative of the position function with respect to time. To find the speed, we first need to find the derivative of $x(t)$ with respect to $t$ , which will give us the velocity $v(t)$ .
Calculate Derivative of Position Function: Calculate the derivative of the position function $x(t) = t^2 + 4t + 15$ . The derivative of $t^2$ is $2t$ , the derivative of $4t$ is $4$ , and the derivative of a constant like $15$ is $0$ . So, $v(t) = \frac{dx}{dt} = 2t + 4$ .
Evaluate Velocity at $t = 4$ : Evaluate the velocity at $t = 4$ . Substitute $t = 4$ into the velocity function $v(t) = 2t + 4$ . $v(4) = 2(4) + 4 = 8 + 4 = 12$ .
Determine Speed at t = $4$ : Since speed is the absolute value of velocity, and the velocity at $t = 4$ is $12$ , the speed is also $12$ , because $12$ is already a positive number. $\newline$ Speed at $t = 4$ is $|v(4)| = |12| = 12$ .