Acceleration of the particle a = (4t² + 5t + 6) m/s². Find the velocity of the particle from 0 to 1 second.

To find the velocity of the particle from an acceleration formula, we integrate the acceleration function. The velocity at 0 seconds is 0 m/s, and at 1 second, it is approximately 9.83 m/s. An initial condition is applied assuming the particle starts from rest.
;

Answered by Anonymous | 2025-07-04

To find the velocity of the particle from 0 to 1 second, when the acceleration is given by a ( t ) = 4 t 2 + 5 t + 6 m/s², we need to integrate the acceleration function with respect to time to find the velocity function.

Acceleration Function :
a ( t ) = 4 t 2 + 5 t + 6

Integrate to Find Velocity :
The velocity function v ( t ) is the integral of the acceleration function:
v ( t ) = ∫ ( 4 t 2 + 5 t + 6 ) d t
Let's integrate:
v ( t ) = ∫ 4 t 2 d t + ∫ 5 t d t + ∫ 6 d t
v ( t ) = 3 4 ​ t 3 + 2 5 ​ t 2 + 6 t + C

Determine the Constant of Integration C :
We need additional information to find the exact value of C . Typically, this is done by knowing the initial velocity. If, for example, the initial condition is that the particle starts from rest ( v ( 0 ) = 0 ), we can use this to solve for C :
v ( 0 ) = 3 4 ​ ( 0 ) 3 + 2 5 ​ ( 0 ) 2 + 6 ( 0 ) + C = 0
Therefore, C = 0 .
Thus, the velocity function is:
v ( t ) = 3 4 ​ t 3 + 2 5 ​ t 2 + 6 t

Find the Velocity from 0 to 1 Second:
Evaluate the velocity at t = 1 second:
v ( 1 ) = 3 4 ​ ( 1 ) 3 + 2 5 ​ ( 1 ) 2 + 6 ( 1 )
v ( 1 ) = 3 4 ​ + 2 5 ​ + 6
Let's calculate this step-by-step:
3 4 ​ ≈ 1.33


2 5 ​ = 2.5
Adding all these together:

[tex]v(1) = 1.33 + 2.5 + 6 = 9.83[/tex] m/s

So, the velocity of the particle at t = 1 second is approximately 9.83 m/s.

Answered by JessicaJessy | 2025-07-06