Question

A horse is at \(x=4.3 \mathrm{~m}\) at \(t=0\) and moves with a constant velocity of \(6.7 \mathrm{~m} / \mathrm{s}\). (a) Plot the motion of the horse for the range of times from \(t=0\) to \(t=5.0 \mathrm{~s}\). (b) Write the corresponding equation of motion.

Short answer

(a) The plot is a straight line from (0, 4.3) to (5, 37.8). (b) The equation of motion is \( x(t) = 4.3 + 6.7t \).

Step-by-step solution

Understanding Constant Velocity Motion

The horse starts at a position of \(x = 4.3\, \text{m}\) at time \(t = 0\, \text{s}\) and moves with a constant velocity of \(v = 6.7\, \text{m/s}\). In constant velocity motion, the position of the object changes linearly with time, which can be described using the kinematic equation: \( x(t) = x_0 + vt \).

Determine the Equation of Motion

To write the equation of motion for the horse, use the initial position \(x_0 = 4.3\, \text{m}\) and the constant velocity \(v = 6.7\, \text{m/s}\). Substitute these values into the kinematic equation:\[ x(t) = 4.3 + 6.7t \]Thus, the equation of motion for the horse is \( x(t) = 4.3 + 6.7t \).

Plotting the Motion From t = 0 to t = 5.0 s

We now plot \( x(t) = 4.3 + 6.7t \) for the range \(t = 0\) to \(t = 5.0\, \text{s}\). To do this, calculate a few key points:- At \(t = 0\, \text{s}\), \(x(0) = 4.3 + 6.7(0) = 4.3\, \text{m}\)- At \(t = 5.0\, \text{s}\), \(x(5.0) = 4.3 + 6.7(5.0) = 4.3 + 33.5 = 37.8\, \text{m}\)Plotting these points on the graph will show a straight line starting from \(4.3\, \text{m}\) at \(t = 0\) and ending at \(37.8\, \text{m}\) at \(t = 5.0\, \text{s}\). The line represents the horse moving with a constant velocity.

Key concepts

kinematic equations

Kinematic equations are powerful tools for analyzing the motion of objects. When dealing with constant velocity motion, one of the simplest forms of these equations can be utilized. This kinematic equation states that the position at any time \(x(t)\) is the initial position \(x_0\) plus the product of velocity \(v\) and time \(t\): \[ x(t) = x_0 + vt \]For our horse, we have an initial position \(x_0 = 4.3\, \text{m}\) and a constant velocity of \(v = 6.7\, \text{m/s}\). By substituting these values into the equation, we obtain the motion equation:\[ x(t) = 4.3 + 6.7t \] This equation elegantly captures how the horse's position changes over time. Each unit of time results in a linear increase in position by the magnitude of the velocity multiplied by time. Kinematic equations like this one are invaluable for predicting future positions and understanding motion under constant velocity.

position vs. time graph

When observing an object's motion, a position vs. time graph provides clear visual insight. In a scenario where the horse moves with a constant velocity, the graph displays a straight line. This reflects the linear relationship between position and time.For the horse initially positioned at \(x = 4.3\, \text{m}\) moving at \(v = 6.7\, \text{m/s}\) over \(t = 0\) to \(5.0\, \text{s}\), the graph gives a clear visual representation:

• At \(t = 0\), the position is \(4.3\, \text{m}\).
• At \(t = 5.0\), the calculated position is \(37.8\, \text{m}\), derived from the equation \(x(5.0) = 4.3 + 6.7 \times 5.0\).

The straight line connecting these points indicates steady, unvarying motion. This representation helps students develop an intuitive understanding of how objects travel in a linear and predictable pattern during constant velocity motion.

linear motion

Linear motion is the simplest form of motion, characterized by a straight path in a specified direction. In this type of motion, objects move at a constant speed along a straight line.

For our horse, this is exhibited by maintaining a constant velocity of \(6.7\, \text{m/s}\). Linear motion involves no change in speed or direction, making it straightforward to model and predict with the kinematic equation.
The horse's movement from \(4.3\, \text{m}\) to \(37.8\, \text{m}\) over a given timeframe results in a direct translation along the x-axis.

What students should take from linear motion is its predictability: once you know the starting point and velocity, calculating future positions becomes a simple task. This makes linear motion a foundational concept in physics, aiding in understanding more complex motions encountered later on.