Final answer:
To determine the initial behavior of a particle's speed with a given velocity and acceleration, we use the dot product of velocity and acceleration. If the result is negative, it indicates that the speed of the particle initially decreases. The correct answer is option (2).
Step-by-step explanation:
The student asked a question regarding a particle being projected with a given initial velocity and a provided acceleration. The task is to determine the initial behavior of the particle's speed.
A particle is projected with a velocity of (10i + 12j) m/s and has a constant acceleration of (3i - 3j) m/s². To determine whether the speed initially increases or decreases, we need to consider the acceleration's components relative to the velocity's components. The acceleration in the x-direction (3i) is in the same direction as the velocity's x-component (10i), which would lead to an increase in the speed in the x-direction. However, the acceleration's y-component (-3j) is in the opposite direction of the velocity's y-component (12j), introducing a deceleration effect in the y-direction. The combined effect of these components must be analyzed to ascertain the overall initial behavior of the particle's speed.
If we analyze the initial speed change by looking at the dot product of velocity and acceleration, we get:
Velocity · Acceleration = (10)(3) + (12)(-3) = 30 - 36 = -6 m²/s³.
Since the dot product is negative, it indicates that the speed of the particle initially decreases. Thus, the correct statement is (2).