Question

Calculate A golfer putts on the eighteenth green at a distance of \(5.0 \mathrm{~m}\) from the hole. The ball rolls straight, in the positive direction, but overshoots the hole by \(1.2 \mathrm{~m}\). The golfer then putts back to the hole and sinks the putt for par. (a) What is the distance traveled by the ball? (b) What is the displacement of the ball?

Short answer

(a) Distance: 7.4 m, (b) Displacement: 0 m.

Step-by-step solution

Understanding the Problem

The problem describes a scenario where a golfer makes two putts. First, the ball travels a distance of 5.0 m to the hole but overshoots by 1.2 m, reaching a total of 6.2 m on the first putt. The second putt is from the overshoot position back to the hole.

Calculating the Distance

Distance is the total path traveled by the ball. For the first putt, it travels 6.2 m (5.0 m to the hole plus 1.2 m overshoot). For the second putt, it travels back 1.2 m to the hole. Thus, the total distance traveled is 6.2 m + 1.2 m.

Distance Formula Application

Add the distances from both putts: \[ 6.2 \, \text{m} + 1.2 \, \text{m} = 7.4 \, \text{m} \]

Distance Traveled by the Ball

The total distance traveled by the ball is 7.4 m.

Calculating the Displacement

Displacement is the straight-line distance from the initial to the final position. Since the ball starts 5.0 m from the hole and ends right at the hole, the displacement is the straight-line length from start to finish.

Displacement Formula Application

Calculate the total displacement: \[ 5.0 \, \text{m} - 5.0 \, \text{m} = 0 \, \text{m} \]

Displacement of the Ball

The displacement of the ball is 0 m, as it ends up at the starting point of the hole.

Key concepts

Physics Problem Solving

Physics problem solving is all about breaking down complex scenarios into understandable parts. It requires a structured approach where you identify given data, understand what is asked, and select appropriate formulas. In our golf problem, we started by understanding the trajectory of the golf ball. This involved identifying the distances moved in each putt. The golfer's actions were simplified into two major movements: the initial putt and the return putt. A consistent approach in physics problem solving involves:

• Identifying known values and units
• Breaking down the problem into smaller steps
• Using proper mathematical formulas and operations
• Verifying if the results make sense in context

By clearly setting out the movements and calculating both total distance and displacement, we were able to solve the problem comprehensively. Understanding the difference between distance and displacement and correctly applying this to the golf putts is key to solving part a) and b) of the exercise.

Linear Motion

In physics, linear motion describes movement along a straight line, which can either be in one direction or back and forth. In the golf problem, the ball experiences linear motion as it moves towards the hole and then overshoots it. It later returns straight back to complete the putt. To better understand, let's consider a few key aspects: The initial movement of the ball took it forward towards the hole, with its path being entirely linear. Since it overshot, the second motion was linear as well, just in the opposite direction. Linear motion is characterized by the simplistic movement without any change in direction or deviation, making calculations easier. Linear motion can be described using:

• Speed: how fast the object moves
• Velocity: the speed with a direction (in our case, forward and backward towards the hole)
• Acceleration: the change in velocity over time

Understanding linear motion helps in comprehending terms like speed, velocity, and the very concepts of distance and displacement. In solving linear motion problems, it’s essential to keep in mind the direction of the motion since displacement considers the straight-line change from start to finish.

Vectors and Scalars

In the world of physics, understanding vectors and scalars is fundamental. These concepts help accurately describe quantities and their properties. Let's break down these terms in the context of our golf ball example: Vectors are quantities that have both a magnitude and a direction. In our exercise, displacement is a vector. The ball's displacement refers to its net change in position from start to finish, which involves direction (from where the golfer stood, to the hole). On the other hand, scalars have only magnitude without direction. Distance traveled in the exercise is a scalar. It doesn't consider the direction, but rather the total length of the path traveled by the golf ball. Important distinctions include:

• Vectors: Displacement, velocity, and acceleration
• Scalars: Distance, speed, and time

In our problem, recognizing that distance involves adding all paths taken and ignoring direction, while displacement considers only the direct route from start to finish, is crucial. These definitions allow for more efficient and accurate calculations when working through physics problems.