A speedboat increases in speed from 21 m/s to 30.9 m/s in a distance 191 m. Find the magnitude of its acceleration.
average speed = (21 +30.9)/2 = 25.95 m/s
so time = 191 m/25.95 m/s = 7.36 seconds
a = change in velocity/change in time
= (30.9-21)/7.36
To find the magnitude of acceleration, we can use the formula:
acceleration (a) = change in velocity (Δv) / change in time (Δt)
In this case, we are given the change in velocity and the distance traveled. We can find the time taken to cover the given distance using the equation of motion:
distance (d) = initial velocity (v0) * t + (0.5) * acceleration (a) * t^2
Here, the initial velocity (v0) is 21 m/s, the final velocity (v) is 30.9 m/s, and the distance (d) is 191 m.
191 m = 21 m/s * t + (0.5) * a * t^2
Simplifying the equation, we get:
0.5 * a * t^2 + 21t - 191 = 0
This is a quadratic equation in terms of time, t. Solving this equation will give us the time taken to cover the given distance. We can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
In this equation, a = 0.5, b = 21, and c = -191.
Substituting these values, we get:
t = (-21 ± √(21^2 - 4 * 0.5 * -191)) / (2 * 0.5)
Solving this equation, we find two possible values for t: 4.46 seconds and -8.56 seconds. We can discard the negative value since time cannot be negative in this context.
Now that we have the time taken (t = 4.46 seconds), we can find the change in velocity (Δv) using the equation:
Δv = final velocity (v) - initial velocity (v0)
Substituting the given values:
Δv = 30.9 m/s - 21 m/s = 9.9 m/s
Finally, we can calculate the magnitude of acceleration (a) using the formula:
a = Δv / t
Substituting the values:
a = 9.9 m/s / 4.46 s ≈ 2.22 m/s^2
Therefore, the magnitude of its acceleration is approximately 2.22 m/s^2.