Final answer:
To determine the velocity at t = 7, we integrated the acceleration function and derived the velocity function. However, with only the position at t = 7 given, we cannot determine the exact velocity from the options provided without additional initial velocity or a different position information.
Step-by-step explanation:
The velocity of a particle at a certain time can be found by integrating the acceleration function with respect to time. Given that the acceleration a(t) = 6t, we can integrate this function to find the velocity function v(t), considering the integration constant to match the position information provided.
By integrating a(t), we find:
v(t) = ∫ a(t) dt = ∫ 6t dt = 3t² + C,
To find C, we use the given position at t = 7, s(7) = 336. We integrate v(t) to get s(t).
s(t) = ∫ v(t) dt = ∫ (3t² + C) dt = t³ + Ct + D
We then solve for constants C and D using initial conditions. However, since only the position at t = 7 is given, we cannot determine these constants directly. We know the velocity at t = 7 is the derivative of the position function at t = 7. Assuming the position function is in the form of s(t) = t³ + Ct (ignoring D as it does not affect the velocity), taking the derivative yields:
s'(t) = 3t² + C, which is the function we got for v(t).
To find v(7), we use the information that at t = 7, s(7) = 336.
V(7) = 3(7)² = 147 m/s
Therefore, the velocity of the particle at t=7 is not one of the provided options. To find the correct answer from the options given, we would need additional information, such as the initial velocity or another position at a different time.