Question

27 sep analyze and interpret data the graph shows data for an object’s position as a function of time. the tangent line at point a has been drawn for you. determine the instantaneous velocity at point a, where t = 2.5 s. in which direction is the object moving? how do you know?

28 sep communicate information the position graph shown represents an object’s motion. write a sentence that describes that motion. by hand or computer, sketch the velocity as a function of time. then act out the motion.

29 sep develop a model the diagram shows a ball rolling at a constant speed along a horizontal track. it comes to a hill and has enough velocity to get over the hill. sketch by hand or computer a dot diagram and a velocity graph for the ball’s motion.

Explanation:

Question 27

Step1: Understand Position - Time Graph

In a position - time graph, the slope of the tangent line at a point gives the instantaneous velocity. The direction of motion is determined by the sign of the slope. If the slope is negative, the object is moving in the negative direction (towards decreasing position), if positive, towards increasing position.

Step2: Analyze Tangent at Point A

Looking at the graph, the tangent line at point A (where \(t = 2.5\space s\)) has a negative slope. This is because as time \(t\) increases (moving to the right on the time axis), the position \(d\) (on the vertical axis) is decreasing.

Step3: Determine Direction

Since the slope of the tangent (instantaneous velocity) is negative, the object is moving in the direction of decreasing position. From the graph's axes, the position is on the \(y -\)axis (labeled \(d(m)\)) and time on the \(x -\)axis (labeled \(t(s)\)). So, as time increases, the position is decreasing, meaning the object is moving towards the origin (or in the direction of decreasing distance from the origin, or in the negative direction of the position axis). We know this because the slope of the tangent to a position - time graph at a point is equal to the instantaneous velocity, and a negative slope implies a negative velocity, which corresponds to motion in the direction of decreasing position.

Step1: Analyze Position - Time Graph

The position - time graph shows that initially, the position of the object is constant (the graph is horizontal), which means the object is at rest. Then, the position starts to decrease, and the slope of the position - time graph (which is velocity) changes. The curve becomes steeper, meaning the velocity (slope) is increasing in magnitude (becoming more negative if we consider the direction of decreasing position) as the object starts to move and speed up towards the direction of decreasing position, until it finally comes to rest (position becomes constant again at the end).

Step2: Describe Motion

The object is initially at rest (constant position over time). Then, it starts to move in the direction of decreasing position, and its speed (magnitude of velocity) increases as it moves until it finally comes to rest (position becomes constant again).

Step3: Sketch Velocity - Time Graph

• For the initial part (constant position), the velocity is \(0\) (horizontal line on velocity - time graph at \(v = 0\)).
• Then, as the position starts to decrease and the slope of the position - time graph becomes more negative (steeper), the velocity (which is the slope) becomes more negative and its magnitude increases. So, the velocity - time graph will show a line (or curve, depending on the exact shape of the position - time graph) that goes from \(0\) to a more negative value, with the slope of the velocity - time graph (acceleration) being constant or changing depending on the curvature of the position - time graph. If the position - time graph is a smooth curve that is getting steeper linearly, the velocity - time graph will be a straight line with negative slope (constant acceleration). If the position - time graph has a non - linear curvature, the velocity - time graph will be a curve. Finally, when the position becomes constant again, the velocity returns to \(0\).

Step1: Analyze Ball's Motion on Horizontal Track

When the ball is rolling on a horizontal track at a constant speed, its velocity is constant (magnitude and direction). So, the dot diagram (where dots represent the position of the ball at equal time intervals) will have dots spaced equally apart (since distance \(d=v\times t\) and \(v\) and \(t\) are constant). The velocity graph will be a horizontal line (constant velocity) at a positive value (assuming the direction of motion is positive) on a velocity - time graph.

Step2: Analyze Ball's Motion on the Hill

When the ball approaches the hill, as it starts to climb the hill, it will slow down (decelerate) because it has to do work against gravity. So, the velocity decreases, but the direction remains the same (still moving forward to get over the hill). The dot diagram will have dots spaced closer together (since \(d = v\times t\) and \(v\) is decreasing, so for the same \(t\), \(d\) is smaller). Then, as it goes down the other side of the hill, it will speed up (accelerate) due to gravity, so the velocity increases back towards its original speed (assuming no energy loss, like friction). The dot diagram will have dots spaced further apart as it speeds up.

Step3: Sketch Dot Diagram

• Horizontal Track (Constant Speed): Draw dots at equal intervals (e.g., dot 1, dot 2, dot 3,... with equal spacing between them) to represent the ball's position at equal time intervals.
• Climbing the Hill (Decelerating): Draw dots with decreasing spacing (dot 4, dot 5, dot 6,... with each subsequent dot closer to the previous one than the dots on the horizontal track) to show the ball slowing down.
• Descending the Hill (Accelerating): Draw dots with increasing spacing (dot 7, dot 8, dot 9,... with each subsequent dot further from the previous one than the dots on the hill - climbing part) to show the ball speeding up. The spacing should approach the spacing of the dots on the horizontal track if it reaches the original speed.

Step4: Sketch Velocity Graph

• Horizontal Track (Constant Speed): Draw a horizontal line at a positive velocity value (\(v_0\)) on a velocity - time graph.
• Climbing the Hill (Decelerating): Draw a line with negative slope (since velocity is decreasing) from \(v_0\) to a minimum velocity \(v_{min}\) (still positive, as it's moving forward to get over the hill).
• Descending the Hill (Accelerating): Draw a line with positive slope (since velocity is increasing) from \(v_{min}\) back to \(v_0\) (assuming no energy loss). If there is energy loss (like friction), it will approach a speed less than \(v_0\).

Answer:

The instantaneous velocity at Point A (\(t = 2.5\space s\)) has a direction of decreasing position (e.g., towards the origin). We know this because the slope of the tangent line to a position - time graph at a point is the instantaneous velocity, and the tangent at Point A has a negative slope, indicating a negative velocity (motion in the direction of decreasing position).

Question 28