Problem 18 - College Algebra and Calculus: An Applied Approach

Question

A racecar travels northward on a straight, level track at a constant speed, traveling \(0.750\) kilometer in \(20.0\) seconds. The return trip over the same track is made in \(25.0\) seconds. (a) What is the average velocity of the car in meters per second for the first leg of the run? (b) What is the average velocity for the total trip?

Step-by-Step Solution

Answer

The average velocity for the first leg of the run is \(37.5 m/s\) and the average velocity for the total trip is \(0 m/s\).

Step 1: Calculate the velocity for the first leg

The average velocity for the first leg of the run is calculated using the formula \(v = \frac{d}{t}\), whereas \(d = 0.75 kms = 750 meters\) and \(t = 20 seconds\). Now input the values for displacement \(d\) and time \(t\) into the formula to calculate velocity. \(v = \frac{750m}{20.0s} = 37.5 \,m/s\)

Step 2: Calculate the total time and displacement for the total trip

For calculating the total velocity, it's necessary to consider the total distance (displacement) and total time spent. The total displacement for the total trip is zero, because the car started and ended at the same position. The total time spent is the sum of times for both trips, i.e. \(20s + 25s = 45s\).

Step 3: Calculate the total average velocity

Average velocity for the total trip can be calculated with the same formula as in step 1. Though remember, for the total trip, the displacement is zero, because the car ended up at the same place it started. Hence \(v = \frac{0m}{45.0s} = 0 m/s\).

Key Concepts

Constant Speed

In the context of physics and motion, constant speed refers to a body moving at an unvarying rate. This means that the object covers equal distances in equal intervals of time, regardless of the length of those time intervals. In the exercise provided, a racecar travels northward on a straight, level track at a constant speed, which simplifies the process of calculating the average velocity because the speed doesn't change during the travel.

When dealing with constant speed, the formula for velocity (\( v = \frac{d}{t} \)) becomes quite straightforward to apply, given that the distance (\( d \) and time (\( t \) can be measured accurately. It's important to note that constant speed only describes the magnitude of the velocity. In physics, velocity is a vector quantity, meaning it has both magnitude and direction, which brings us to the concept of displacement.

Displacement

Displacement is a vector quantity that refers to the change in position of an object. It is not the same as distance traveled; instead, it measures the shortest path between the starting and ending point in a straight line, and it includes direction. In the exercise, the racecar returns to its original position after the second leg of the trip, leading to a displacement of zero since there has been no change in position from the start to the end of the complete journey.

When calculating the average velocity across the entire trip, displacement is crucial. Since displacement for the entire journey is zero—because the racecar has returned to its starting point—the average velocity for the entire trip is also zero, regardless of the time taken. This example demonstrates a scenario where understanding the difference between distance traveled and displacement can impact the calculations dramatically.

Average Velocity Formula

The average velocity formula is a fundamental concept in kinematics, represented by \( v = \frac{d}{t} \) where \( v \) stands for average velocity, \( d \) represents displacement, and \( t \) denotes the time interval. To calculate average velocity, you essentially divide the total displacement by the total time taken.

In the case of the racecar scenario, the calculation of average velocity for the first leg was straightforward: the displacement was 750 meters northward, and the time taken was 20 seconds. Therefore, using the average velocity formula, the average velocity equated to \( 37.5 \, m/s \) northward. However, the average velocity for the complete trip is zero because the displacement is zero—the start and end points are the same, showcasing the difference between average speed and average velocity in scenarios involving direction changes.