In 45 seconds, a rocket goes from traveling at [tex]$75 m / s$[/tex] to [tex]$570 m / s$[/tex]. What is the acceleration of the rocket?
[tex]$a=[?] \frac{m}{s^2}$[/tex]
Enter the sign (+ or -) and the magnitude.
Remember: [tex]$a =\frac{\Delta V }{ t }$[/tex]
Answer (2)
Calculate the change in velocity: Δ v = v f − v i = 570 m / s − 75 m / s = 495 m / s .
Apply the acceleration formula: a = t Δ v = 45 s 495 m / s .
Calculate the acceleration: a = 11 s 2 m .
The acceleration of the rocket is 11 s 2 m .
Explanation
Problem Analysis We are given the initial velocity, final velocity, and the time interval for a rocket's acceleration. We need to find the acceleration of the rocket.
Formula for Acceleration The formula for acceleration is given by: a = t Δ v = t v f − v i where:
a is the acceleration,
Δ v is the change in velocity,
v f is the final velocity,
v i is the initial velocity,
t is the time interval.
Given Values We are given:
Initial velocity, v i = 75 s m
Final velocity, v f = 570 s m
Time interval, t = 45 s
Calculating Acceleration Now, we plug in the given values into the formula: a = 45 s 570 s m − 75 s m = 45 s 495 s m = 11 s 2 m Since the rocket is speeding up, the acceleration is positive.
Final Answer The acceleration of the rocket is 11 s 2 m .
Examples
Understanding acceleration is crucial in many real-world scenarios, such as designing vehicles, analyzing sports performance, and even understanding the motion of celestial bodies. For example, when designing a car, engineers need to calculate the acceleration required to reach a certain speed within a specific time frame. Similarly, in sports, the acceleration of a sprinter determines how quickly they can reach their top speed. In space exploration, understanding the acceleration of rockets is vital for planning trajectories and reaching destinations efficiently. The formula a = t Δ v is a fundamental concept that helps us quantify and analyze these scenarios.
The acceleration of the rocket is calculated using the change in velocity over time, resulting in an acceleration of 11 s 2 m . This indicates that the rocket is speeding up. Since the value is positive, it confirms the rocket's increase in speed.
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