Question

Calculate A billiard ball travels \(22 \mathrm{~cm}\) in the positive direction, hits the cushion and rebounds in the negative direction, and finally comes to rest \(7.5 \mathrm{~cm}\) behind its original position. (a) What is the distance covered by the ball? (b) What is the displacement of the ball?

Short answer

(a) Distance: 51.5 cm (b) Displacement: -7.5 cm

Step-by-step solution

Understanding Distance

Distance is the total length of the path traveled by the object, regardless of the direction. In this problem, the ball moves 22 cm to the positive direction first. Then it returns and moves an additional 7.5 cm in the negative direction past its original starting point. To calculate the total distance covered, we add both segments: 22 cm and (7.5 + 22) cm.

Calculating Total Distance

In this step, we calculate the total distance covered by the ball. First, it travels 22 cm, and after rebounding, it travels an additional 7.5 cm beyond the starting point. So, the rebound path is 22 cm (to the cushion) + 7.5 cm = 29.5 cm. The total distance is then:\[ 22 \text{ cm (forward)} + 29.5 \text{ cm (backward)} = 51.5 \text{ cm} \]

Understanding Displacement

Displacement is the change in position of the object, measured as a straight line from the starting point to the final position, and it includes direction. Here, the ball ends up 7.5 cm behind its original position in the negative direction. Thus, the displacement is simply -7.5 cm.

Calculating Displacement

The displacement, which includes direction, is calculated by comparing initial and final positions. The ball started at 0 and ended at -7.5 cm, hence:\[ ext{Displacement} = -7.5 \text{ cm} \] It indicates a movement of 7.5 cm in the negative direction.

Key concepts

Billiard Ball Motion

When we talk about the motion of a billiard ball on a table, we imagine how it travels along the surface. The key elements of this movement involve the various paths taken by the ball. Initially, the ball moves 22 cm in one direction. Once it hits the cushion, it rebounds and continues to move past its starting point. The entire process includes moving forward, hitting an obstacle, and then changing direction. Understanding these movements helps us calculate both the distance traveled and the displacement. Billiard ball motion embraces the concept of linear motion, which involves traveling along a straight path on the pool table.

Positive and Negative Directions

In physics, direction matters a lot, especially when describing motion. For a billiard ball, moving 'forward' could be considered the positive direction, while 'backward' could be the negative direction. These directions are arbitrary but necessary. When the ball moves the initial 22 cm, that's positive. When it rebounds and moves further to end 7.5 cm behind its starting point, that's negative. Understanding these directions is crucial when discussing displacement, as the final position relative to the starting point dictates whether it's positive or negative. This directional assignment helps us understand and solve many physics problems easily.

Calculating Distance

Distance is the total length of the path traveled by an object, without considering the direction. It tells us how much ground the object covered, regardless of its starting or ending point. In the case of the billiard ball, calculating distance involves adding all segments of its travel. The ball first moves 22 cm forward, reaching the cushion. Then, it moves back 22 cm to the starting point and continues 7.5 cm more. The total distance the ball covered is calculated as follows:

• Forward: 22 cm
• Backward: 22 cm + 7.5 cm = 29.5 cm

The sum of distances is therefore:
\[ 22 ext{ cm} + 29.5 ext{ cm} = 51.5 ext{ cm} \] This gives us the total travel path of the ball.

Calculating Displacement

While distance focuses on total path length, displacement is about change in position, with direction. Displacement measures the straight line distance from the initial position to the final position. Consider it as a shortcut from origin to destination. In the billiard ball scenario, it starts at 0 cm and ends 7.5 cm behind the start, giving us a displacement:
\[ ext{Displacement} = ext{Final Position} - ext{Initial Position} \] Which translates to:
\[ ext{Displacement} = -7.5 ext{ cm} - 0 ext{ cm} = -7.5 ext{ cm} \] Negative indicates direction behind the starting point. Thus, while the physical movement exceeds 50 cm, the displacement shows a simple backward step of 7.5 cm.