The relative position displacement of an object (in meters) is 2t^2 +6t. Calculate its velocity as t = 4 seconds.

To find the velocity of an object given its relative position displacement, we need to differentiate the displacement function with respect to time. The displacement function you've provided is $s(t)=2t^2+6t$. Let's go through the steps to calculate the velocity at $t=4$ seconds.

Step 1: Differentiate the Displacement Function

The velocity $v(t)$ is the first derivative of the displacement function $s(t)$. We can apply basic differentiation rules here:

• The derivative of $t^n$ is $n\cdot t^{n-1}$.
• The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

Applying these rules, we differentiate $s(t)$:

$s(t)=2t^2+6t$

$v(t)=\frac{ds}{dt}=\frac{d}{dt}(2t^2)+\frac{d}{dt}(6t)$

$v(t)=2\cdot 2t^{2-1}+6\cdot 1=4t+6$

Step 2: Substitute $t=4$ into the Velocity Function

Now that we have the velocity function $v(t)=4t+6$, we can find the velocity at $t=4$ seconds:

$v(4)=4(4)+6$

$v(4)=16+6=22$

Final Result

Thus, the velocity of the object at $t=4$ seconds is $22$ meters per second. This means that at that specific moment, the object is moving forward at a speed of 22 m/s.

Understanding the Concept

Velocity is a vector quantity that indicates both the speed and direction of an object's motion. In this case, since we only calculated the magnitude, we can say the object is moving in the positive direction at that speed. If you have any further questions about displacement, velocity, or related concepts, feel free to ask!