Question

The acceleration of a particle is given by a(t) = 6t. At time t = 0, the position of the particle is s(t) = 0. When t = 7, the position is s(t) = 336. What is the velocity of the particle at time t = 7?

1) 42
2) 48
3) 54
4) 60

Answer 1

Integrating the provided acceleration function and applying the initial condition results in a calculated velocity at t = 7 that is not reflected in the given options, indicating a need to re-assess the information and calculations.

Step-by-step explanation:

To find the velocity of the particle at time t = 7 when the given acceleration a(t) = 6t, we can integrate the acceleration function to get the velocity function v(t). Since acceleration is the time derivative of velocity, we integrate a(t) with respect to time to get v(t) = ∫ a(t) dt = ∫ 6t dt = 3t² + C, where C is the constant of integration. To determine C, we use the initial condition that at t = 0, the position s(t) = 0; this implies an initial velocity of v(0) = 0. This gives us C = 0, and thus v(t) = 3t². Now, to find the velocity at t = 7, we simply substitute 7 into our velocity function, yielding v(7) = 3*(7²) = 3*49 = 147 m/s, which is not one of the provided options. Hence, it appears there is a discrepancy in the provided information, and we must re-evaluate our given data, initial conditions, and steps to ensure accuracy before providing a final answer.