Suppose a particle travels along the x-axis from (-14m, 0) to (15m, 0), and then to (-8m, 0) in 20 seconds.
a) Find the total distance travelled and displacement of the particle.
b) Calculate the average speed and average velocity.
Answer (2)
Calculate the total distance by summing the distances between each point: d = ∣15 − ( − 14 ) ∣ + ∣ − 8 − 15∣ = 29 + 23 = 52 m .
Determine the displacement by finding the difference between the final and initial positions: Δ x = − 8 − ( − 14 ) = 6 m .
Compute the average speed by dividing the total distance by the total time: v a vg = 20 52 = 2.6 m/s .
Calculate the average velocity by dividing the displacement by the total time: v ˉ = 20 6 = 0.3 m/s .
d = 52 m , Δ x = 6 m , v a vg = 2.6 m/s , v ˉ = 0.3 m/s
Explanation
Problem Analysis We are given the initial position, an intermediate position, the final position, and the total time taken for a particle's journey along the x-axis. We need to find the total distance traveled, the displacement, the average speed, and the average velocity.
Calculating Distances First, we calculate the distance d 1 from the initial position x i = − 14 m to the intermediate position x mi d = 15 m: d 1 = ∣ x mi d − x i ∣ = ∣15 − ( − 14 ) ∣ = ∣15 + 14∣ = ∣29∣ = 29 m Then, we calculate the distance d 2 from the intermediate position x mi d = 15 m to the final position x f = − 8 m: d 2 = ∣ x f − x mi d ∣ = ∣ − 8 − 15∣ = ∣ − 23∣ = 23 m
Total Distance and Displacement The total distance d traveled is the sum of d 1 and d 2 :
d = d 1 + d 2 = 29 + 23 = 52 m The displacement Δ x is the difference between the final and initial positions: Δ x = x f − x i = − 8 − ( − 14 ) = − 8 + 14 = 6 m
Average Speed and Velocity Next, we calculate the average speed v a vg using the total distance and the total time t = 20 s: v a vg = t d = 20 s 52 m = 2.6 m/s The average velocity v ˉ is calculated using the displacement and the total time: v ˉ = t Δ x = 20 s 6 m = 0.3 m/s
Final Answer Therefore, the total distance traveled is 52 m, the displacement is 6 m, the average speed is 2.6 m/s, and the average velocity is 0.3 m/s.
Examples
Understanding displacement, distance, speed and velocity is crucial in many real-world applications. For example, when planning a road trip, knowing the total distance helps estimate fuel consumption, while understanding displacement helps determine the most direct route. Similarly, in sports, analyzing an athlete's average speed and velocity can provide insights into their performance and efficiency during a race or game. These concepts are fundamental in physics and engineering for analyzing motion and optimizing performance.
The total distance traveled by the particle is 52 m, while the displacement is 6 m. The average speed of the particle is 2.6 m/s and the average velocity is 0.3 m/s.
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