Problem 18 - Calculus an Applied Approach

Question

Physics: Velocity 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. The average velocity for the entire trip is 33.33 m/s.

Step 1: Convert distance from kilometers to meters

The car travels 0.750 kilometers. To convert kilometers to meters (because we want the velocity in meters per second), we multiply 0.750 by 1000, which gives us 750 meters.

Step 2: Calculate Velocity For First Leg Of The Run

The formula for velocity is distance/time. The car travels 750 meters in 20 seconds on the first leg of the run, so the average velocity during this segment is \( \frac{750m}{20s} = 37.5 m/s \)

Step 3: Calculate Total Distance And Total Time

The total distance travelled by the car is twice the distance of one leg, because it goes there and back. So, the total distance is \(2 * 750m = 1500m\). Similarly, the total time is the time of the first leg plus the time of the second leg, which is \(20s + 25s = 45s\).

Step 4: Calculate Velocity For The Total Trip

Now, use the formula for velocity with these total values. The average velocity for the entire trip is \( \frac{1500m}{45s} = 33.33 m/s \)

Key Concepts

Physics Velocity Problems

Understanding velocity in physics involves analyzing motion and the speed at which an object covers a distance in a certain direction. It's a vector quantity, characterized by both magnitude and direction, distinguishing it from speed, which lacks directional information.

In physics problems, velocity can change due to acceleration, or remain constant in uniform motion. Students frequently encounter scenarios needing calculations for initial velocity, final velocity, average velocity, or the effects of different forces on an object's motion. In the example of the racecar, the problem is simplified as the velocity is constant, allowing for a straightforward application of the average velocity formula.

Converting Kilometers to Meters

In physics and many other scientific areas, converting between different units of measurement is a fundamental skill. Metric conversions are crucial as most scientific measurements use the International System of Units (SI). For instance, when calculating velocity, it's often required to convert distance from kilometers to meters for coherence with speed measured in meters per second.

To convert kilometers to meters, you simply multiply the distance in kilometers by 1,000, since one kilometer is equivalent to 1,000 meters. This allows for uniform units when applying formulas, making calculations cleaner and reducing the risk of mistakes due to unit inconsistency.

Velocity Formula Application

The velocity of an object is calculated by dividing the distance traveled by the time it takes to travel that distance. The formula is represented by the equation: \( v = \frac{d}{t} \) where \( v \) represents velocity, \( d \) stands for distance, and \( t \) denotes time. In the case of uniform motion, like the racecar's first leg, applying this formula straightforwardly yields the average velocity. This direct application aids in understanding both uniform motion and the general relationship between velocity, distance, and time.

Calculating Distance and Time

In any physics problem involving motion, accurately determining distance and time is essential. Distance is the total path length traveled, and time is the duration of travel. When calculating average velocity for a round trip, it's crucial to combine the distances and times for each leg of the journey.

In the racecar example, the total distance is the sum of both the northward and return legs, while total time is the sum of the time taken for each leg. Understanding how to properly tally these values is vital for accurate velocity calculations and a firm grasp on how motion unfolds over intervals and through spaces.