Problem 71 - College Physics

Question

A person riding in the back of a pickup truck traveling at \(70 \mathrm{~km} / \mathrm{h}\) on a straight, level road throws a ball with a speed of \(15 \mathrm{~km} / \mathrm{h}\) relative to the truck in the direction opposite to the truck's motion. What is the velocity of the ball (a) relative to a stationary observer by the side of the road, and (b) relative to the driver of a car moving in the same direction as the truck at a speed of \(90 \mathrm{~km} / \mathrm{h} ?\)

Step-by-Step Solution

Answer

(a) 55 km/h; (b) -35 km/h relative to the car.

Step 1: Determine the velocity relative to a stationary observer

First, identify the velocity of the ball relative to the stationary observer. The truck is moving at \(70 \, \text{km/h}\) and the ball is thrown at \(15 \, \text{km/h}\) in the opposite direction, making the velocity of the ball relative to a stationary observer \(70 \, \text{km/h} - 15 \, \text{km/h} = 55 \, \text{km/h}\). Therefore, the velocity of the ball relative to the stationary observer is \(55 \, \text{km/h}\) in the direction of the truck.

Step 2: Determine the velocity relative to a moving observer

Next, calculate the velocity of the ball relative to a driver moving with the car at \(90 \, \text{km/h}\) in the same direction as the truck. The velocity of the ball relative to the stationary observer is \(55 \, \text{km/h}\). Hence, the relative velocity between the ball and the moving car observer is \(55 \, \text{km/h} - 90 \, \text{km/h} = -35 \, \text{km/h}\). This indicates the ball is moving at \(35 \, \text{km/h}\) in the opposite direction of the car.

Key Concepts

pickup truck motion

When understanding the concept of pickup truck motion, it's key to think about how the truck is moving along the road. In our scenario, the truck is traveling at a steady speed of \(70 \text{ km/h}\).
This constant speed in a straight line sets the basis for calculating other motions around it, such as the motion of a ball inside the truck or in relation to other objects or observers.

Here are a few reasons why understanding the pickup truck's motion is important:

• The truck’s motion serves as a reference point for other calculations.
• Knowing the truck’s speed helps us understand the influence it has on anything that moves inside or is associated with it, such as a person throwing a ball.
• The truck's motion is essential when comparing it to other moving or stationary objects to determine relative velocities.

Overall, the pickup truck's motion is the foundation for understanding the relative velocities in the exercise.

stationary observer

A stationary observer refers to someone or something that does not move. In the context of this problem, imagine someone standing on the side of the road, watching the truck and the ball. This observer provides a fixed point of view to compare speeds.

To find the velocity of the ball relative to this stationary observer, we need to examine how the ball’s speed combines with or opposes the truck’s speed. In our case, the truck moves at \(70 \text{ km/h}\), and the ball is thrown at \(15 \text{ km/h}\) in the opposite direction. This results in a speed of \(55 \text{ km/h}\) for the ball from the point of view of the stationary observer.

Here's a quick breakdown:

• The stationary observer sees the truck moving forward at \(70 \text{ km/h}\).
• The ball’s speed of \(15 \text{ km/h}\) is subtracted from the truck's speed because it is thrown backward.
• Thus, the ball appears to move at \(55 \text{ km/h}\) in the direction the truck is moving.

Understanding the perspective of a stationary observer gives clarity to calculating speeds that differ from moving reference points.

moving observer

A moving observer is someone who is also in motion relative to the object being observed. In this problem, the moving observer is the driver of a car traveling in the same direction as the pickup truck but at a speed of \(90 \text{ km/h}\).
To find the ball's velocity relative to this moving observer, we need to consider the speeds of the truck, the ball, and the car.

Here’s how it breaks down:

• The ball is perceived to be moving at \(55 \text{ km/h}\) based on the stationary observer's perspective, as calculated earlier.
• The car is going faster than the speed of the ball as seen by the stationary observer.
• Because the car is moving faster, to the car driver, the ball appears to be moving backward at \(35 \text{ km/h}\) after calculating \(55 \text{ km/h} - 90 \text{ km/h} = -35 \text{ km/h}\).

This scenario illustrates how the motion of objects can appear differently depending on the observer's motion, providing insight into the differing frames of reference.

velocity calculation

Velocity calculation in physics involves determining the speed and direction of one object relative to another. In problems like this, where multiple observers are involved, understanding relative velocity is crucial.

To calculate relative velocity, consider these factors:

• Identify the direction of each object’s movement.
• Take note if they move in the same direction or opposite directions.
• Use basic arithmetic to add or subtract velocities accordingly.
  For instance, the velocity of the ball with respect to the stationary observer requires subtracting the thrown ball's speed from the truck's speed since they move in opposite directions.
• With a moving observer, calculate how quickly one object overtakes or lags behind another, such as the faster-moving car in the problem.

This fundamental understanding of velocity calculation helps to correctly interpret motion from various perspectives, ensuring each observer sees the speed and direction appropriately.