Question

Which one of the following statements is true at \(t=4\) 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) 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. Answer: Insufficient information is given about the object's motion to determine which statement is true at t = 4 seconds.

Step-by-step solution

Understanding the given statements

We have the following statements given in the question: 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. To choose the correct statement at t = 4s, we need more information about the object's motion, such as its position, velocity, and acceleration functions. Without any given information about the object's motion (position, velocity, or acceleration functions), we cannot determine which statement is true at t = 4 s.

Key concepts

Velocity Components

Velocity is a vector quantity that describes both speed and direction of motion. In physics, we often decompose this vector into perpendicular components, typically along the x and y axes in a plane. The velocity components of an object tell us how fast the object is moving along each axis. For instance, if an object is moving northeast at a constant rate, its x-component of velocity would be positive (eastward), and its y-component of velocity would also be positive (northward).

Understanding velocity components is crucial because it allows us to predict the object's position at any given time and to analyze the effects of forces acting on the object. When we say the x-component of the velocity is zero, as in option a), it means that there is no motion along the horizontal axis at that moment. However, it does not necessarily mean the object is not moving — the object could still have a non-zero velocity in the y-component, indicating upward or downward motion.

Acceleration Components

Similar to velocity, acceleration is also a vector and can be broken down into components along the axes. The acceleration components tell us how the velocity of the object is changing over time in each direction. If an object is speeding up in a straight line, its acceleration component in the direction of motion is positive. Conversely, if it's slowing down, the acceleration component in the direction of motion is negative. When an object is moving in a curve, the acceleration components can reveal how it is speeding up or slowing down, as well as changing direction.

In the context of the original problem, deciding whether statement b) The x-component of the acceleration of the object is zero is true would require data about the forces acting on the object or the change in the velocity over time. Zero acceleration along the x-axis would mean that the object's velocity in that direction is not changing; it could be moving at a constant speed or not moving at all along the x-axis.

Object Motion Analysis

In object motion analysis, we use physics principles to describe and predict the motion of objects. Kinematic equations, velocity components, and acceleration components are tools that help in this analysis. By understanding the initial position, initial velocity, and acceleration of an object, we can determine its future position and velocity at any given time.

To analyze an object's motion, we first need to establish a reference frame and coordinate system. With that in place, we can monitor how its position changes over time, which gives us its velocity, and how its velocity changes over time, which gives us its acceleration. Consistent with the step-by-step solution provided, more information about the object's motion is necessary to make any conclusive statement about the motion at any specific time, such as t = 4 seconds.

Kinematic Equations

The kinematic equations are a set of formulas used to solve problems involving the motion of objects without the influence of forces (except for gravity, when considering vertical motion). These equations relate the five kinematic variables: displacement, initial velocity, final velocity, acceleration, and time.

Common Kinematic Equations Include:

• \( v = v_0 + at \) (final velocity)
• \( s = s_0 + v_0 t + \frac{1}{2}at^2 \) (displacement)
• \( v^2 = v_0^2 + 2a(s - s_0) \) (relationship between velocity and displacement)
• \( s = v_0 t + \frac{1}{2}at^2 \) (displacement without initial displacement)

These equations are essential in determining an object's position and velocity at any point in time. They are particularly useful when one of the variables is unknown and we need to solve for it using the others. For the original exercise, given additional information such as initial conditions and any constant acceleration, one could use these equations to determine the correct answer to the problem at the specified time.