$s(t)$ is the position of a body moving along a coordinate line; $s(t)$ is measured in feet and $t$ in seconds, where $t \geq 0$. Find the position, velocity, and speed of the body at the indicated time. State your solution as a decimal rounded to the nearest hundredth.
$s(t)=\frac{4 t}{t^2+1} ; \quad T=2$
POSITION: $\square$ FT
VELOCITY: $\square$ FT/SEC
SPEED: $\square$ FT/SEC
Answer (2)
Position at t = 2 : Substitute t = 2 into s ( t ) to get s ( 2 ) = 2 2 + 1 4 ( 2 ) = 1.6 .
Velocity at t = 2 : Find the derivative v ( t ) = s ′ ( t ) = ( t 2 + 1 ) 2 4 − 4 t 2 , then substitute t = 2 to get v ( 2 ) = − 0.48 .
Speed at t = 2 : Take the absolute value of the velocity to get speed ( 2 ) = ∣ v ( 2 ) ∣ = 0.48 .
The position, velocity, and speed at t = 2 are 1.6 ft, − 0.48 ft/sec, and 0.48 ft/sec, respectively.
Explanation
Problem Setup We are given the position function s ( t ) = f r a c 4 t t 2 + 1 and we need to find the position, velocity, and speed at t = 2 .
Finding Position First, let's find the position at t = 2 by substituting t = 2 into the position function: s ( 2 ) = 2 2 + 1 4 ( 2 ) = 4 + 1 8 = 5 8 = 1.6 So, the position at t = 2 is 1.6 feet.
Finding Velocity Next, we need to find the velocity function v ( t ) . Velocity is the derivative of the position function with respect to time. So, we need to find s ′ ( t ) . Using the quotient rule, we have: v ( t ) = s ′ ( t ) = d t d ( t 2 + 1 4 t ) = ( t 2 + 1 ) 2 ( t 2 + 1 ) ( 4 ) − ( 4 t ) ( 2 t ) = ( t 2 + 1 ) 2 4 t 2 + 4 − 8 t 2 = ( t 2 + 1 ) 2 4 − 4 t 2 Now, we evaluate the velocity at t = 2 :
v ( 2 ) = ( 2 2 + 1 ) 2 4 − 4 ( 2 ) 2 = ( 4 + 1 ) 2 4 − 16 = 25 − 12 = − 0.48 So, the velocity at t = 2 is − 0.48 ft/sec.
Finding Speed Now, we need to find the speed at t = 2 . Speed is the absolute value of the velocity. So, speed ( 2 ) = ∣ v ( 2 ) ∣ = ∣ − 0.48∣ = 0.48 So, the speed at t = 2 is 0.48 ft/sec.
Final Answer Therefore, the position at t = 2 is 1.6 ft, the velocity at t = 2 is − 0.48 ft/sec, and the speed at t = 2 is 0.48 ft/sec.
Examples
Understanding position, velocity, and speed is crucial in many real-world applications. For example, in sports, analyzing the motion of a ball or an athlete helps improve performance. In traffic engineering, knowing the velocity and acceleration of vehicles helps design safer roads and traffic control systems. In robotics, controlling the position and velocity of robot arms is essential for precise manufacturing and assembly.
At time t = 2 , the body's position is 1.6 ft, its velocity is -0.48 ft/sec, and its speed is 0.48 ft/sec. The position is calculated by evaluating the position function, the velocity by differentiating the position function, and speed as the absolute value of velocity.
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