Question

A body moves with its position varying as R = (4t^2−16t)i + 3t^2j meters. What is the y component of its velocity when the x component is zero? Find also the position vector of the body at this instant.

y component = 24t - 24, position vector = (-64, 0)
y component = 6t^2 - 16t, position vector = (0, -64)
y component = 24t - 16, position vector = (0, -64)
y component = 6t^2 - 24t, position vector = (-64, 0)

Answer 1

The y component of velocity when the x component is zero is 12 m/s, and the position vector at this time is (0, 12) meters.

Step-by-step explanation:

The student wants to know the y component of the velocity when the x component is zero, given a position function R = (4t^2−16t)i + 3t^2j meters. To find the velocity, we take the derivative of the position function with respect to time, which gives us V(t) = vx(t)î + vy(t)Ƶ. The x component of the velocity is derived from the x component of the position function, vx(t) = d(4t^2 - 16t)/dt = 8t - 16.

When the x component of velocity, vx(t), is zero, we solve for t. Equating 8t - 16 to zero yields t = 2 seconds. Now we find the y component of velocity: vy(t) = d(3t^2)/dt = 6t. At t = 2 seconds, the y component of velocity is 6(2) = 12 meters per second. The position vector at t = 2 seconds is obtained by substituting t back into the position function, yielding R = (0)i + (12)j meters. Therefore, the correct y component of its velocity when the x component is zero is 12 m/s, and the position vector at this instant is (0, 12).