A particle moves along a straight line relative to a fixed origin $O$ with acceleration, measured in $ft / s ^2$, given by the function $a(t)=20 t-12 t^2$, where $t$ is the time in seconds. If the particle has velocity $v=8 ft / s$ when $t=1 s$, calculate the speed of the particle at the moment $t=2 s$.
a $30 ft / s$
b $10 ft / s$
c $12 ft / s$
d $8 ft / s$
e $2 ft / s

Integrate the acceleration function a ( t ) = 20 t − 12 t 2 to find the velocity function v ( t ) = 10 t 2 − 4 t 3 + C .
Use the initial condition v ( 1 ) = 8 to solve for the constant of integration C , which gives C = 2 .
The velocity function is v ( t ) = 10 t 2 − 4 t 3 + 2 .
Evaluate the velocity at t = 2 to find the speed: v ( 2 ) = 10 ( 2 ) 2 − 4 ( 2 ) 3 + 2 = 10 . The speed is 10 ​ ft/s.

Explanation

Problem Setup We are given the acceleration function a ( t ) = 20 t − 12 t 2 and the initial velocity v ( 1 ) = 8 ft/s. We need to find the velocity at t = 2 seconds.

Finding the Velocity Function First, we need to find the velocity function v ( t ) by integrating the acceleration function a ( t ) with respect to time t . So, we have


$v(t) = ∫ a ( t ) d t = ∫ ( 20 t − 12 t 2 ) d t

Integrating Acceleration Evaluating the indefinite integral, we get

v ( t ) = 10 t 2 − 4 t 3 + C
where C is the constant of integration.

Using Initial Condition Now, we use the initial condition v ( 1 ) = 8 to find the value of C . Plugging in t = 1 into the velocity function, we have

8 = 10 ( 1 ) 2 − 4 ( 1 ) 3 + C

Solving for C Solving for C , we get

8 = 10 − 4 + C 8 = 6 + C C = 2

Complete Velocity Function Thus, the complete velocity function is

v ( t ) = 10 t 2 − 4 t 3 + 2

Calculating Velocity at t=2 Now, we calculate the velocity at t = 2 seconds. Plugging in t = 2 into the velocity function, we have

v ( 2 ) = 10 ( 2 ) 2 − 4 ( 2 ) 3 + 2

Evaluating v(2) Evaluating v ( 2 ) , we get

v ( 2 ) = 10 ( 4 ) − 4 ( 8 ) + 2 = 40 − 32 + 2 = 10
The velocity of the particle at t = 2 is 10 ft/s.

Finding the Speed The speed of the particle at t = 2 is the absolute value of the velocity, which is ∣10∣ = 10 ft/s.

Examples
Understanding motion is crucial in many fields. For example, when designing a roller coaster, engineers use calculus to model the acceleration and velocity of the cars. By knowing the acceleration function, they can integrate it to find the velocity function and ensure the ride is both thrilling and safe. Similarly, in robotics, controlling the motion of robotic arms requires precise calculations of velocity and acceleration to perform tasks accurately.

Answered by GinnyAnswer | 2025-07-08

The speed of the particle at t = 2 seconds is calculated to be 10 ft/s after integrating the acceleration function and using the given initial conditions. Therefore, the correct choice is option (b) . This result is derived through the integration process and evaluating the velocity function at the specified time.
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Answered by Anonymous | 2025-07-16