Question

Which one of the following statements is true at \(t=1\) s? a) The \(x\) -component of the velocity of the object is zero. b) The \(x\) -component of the acceleration of the object is zero. c) The \(x\) -component of the velocity of the object is positive. d) The \(x\) -component of the velocity of the object is negative.

Short answer

A. Velocity component is zero. B. Acceleration component is zero. C. Velocity component is positive. D. Velocity component is negative. Answer: Without any further information about the motion of the object, it is impossible to determine which of the statements is true. However, we have explained the implications and situations when each statement might be true.

Step-by-step solution

Statement A: Velocity Component is Zero

This statement suggests that at \(t=1\) s, the \(x\)-component of the object's velocity is zero. This could be true in cases where the object is momentarily stationary in the \(x\) direction or changing direction along the motion in the x-axis.

Statement B: Acceleration Component is Zero

This statement implies that at \(t=1\) s, the \(x\)-component of the object's acceleration is zero. This could be true if the object is moving with a constant velocity in the \(x\) direction or if the net force on the object is zero in the \(x\) direction.

Statement C: Velocity Component is Positive

This statement suggests that at \(t=1\) s, the \(x\)-component of the object's velocity is positive. This could be true if the object is moving in the positive \(x\) direction at \(t=1\) s.

Statement D: Velocity Component is Negative

This statement implies that at \(t=1\) s, the \(x\)-component of the object's velocity is negative. This could be true if the object is moving in the negative \(x\) direction at \(t=1\) s. Without any further information about the motion of the object, it is impossible to determine which of the statements is true. However, we have explained the implications and situations when each statement might be true.

Key concepts

Velocity Component

Understanding the velocity of an object is crucial when analyzing motion. In physics, velocity is not just about how fast something moves; it's also about the direction of movement. This directional aspect is why we speak of 'velocity components'. The velocity at any given time can be broken down into components along different axes — typically the x-axis (horizontal) and the y-axis (vertical).

For instance, if an object is moving diagonally upward to the right, it has positive components in both the x (rightward motion) and y (upward motion) directions. Alternatively, if it's moving diagonally downward to the left, it would have a negative x-component and a negative y-component.

A critical concept to grasp is that velocity can change over time, which brings us to the next core idea: acceleration.

Acceleration Component

Acceleration, much like velocity, can be dissected into components along the axes of a coordinate system. However, while velocity deals with the rate of change of position, acceleration is the rate at which velocity itself changes. It's an important distinction meaning that acceleration can occur even when an object isn't speeding up or slowing down, but simply changing direction.

Now, a zero acceleration component in a particular direction, say along the x-axis, means that there's no change in the velocity's magnitude or direction in that axis; this occurs when an object moves at a constant velocity or is at rest in relation to the x-axis. Acceleration can also be a result of net forces acting on an object, which links to Newton's Second Law of Motion, our next topic.

Motion Analysis

When one carries out motion analysis, it goes beyond simply stating the current position or velocity of an object. It involves looking at the entire trajectory of motion — where the object started, where it is at every moment, where it's heading, and how its speed and direction change with time.

In our context, analyzing the motion at time 't=1 s' would involve considering all these aspects. For complete motion analysis, one would collect data about the object's position, velocity, acceleration, and forces at play over time. The gathered data then inform us about the patterns and predict future motion under similar conditions. This comprehensive approach is indispensable for understanding complex motions like projectiles or objects under the influence of multiple forces.

Net Force

Net force is the overall force acting on an object when all individual forces are considered together. It's a vector sum of all the forces, and it determines the object's acceleration due to Newton's Second Law of Motion, which states that force equals mass times acceleration (\( F = m \times a \)).

Identifying the net force helps us understand motion changes, such as when an object starts, stops, or changes direction. A zero net force in any given direction, say the x-axis, suggests the object is either at rest or moving with a constant velocity in that direction because no force is causing a change in its motion. Therefore, to cause acceleration, there must be a non-zero net force acting on the object.