an object moves along x axis such that its velocity varies with time as v=(50-10t) where t is time in seconds average speed of the particle in the time interval t=0 to t=6s is equal to
Well, let's calculate the average speed of this object from t=0 to t=6s.
To find average speed, we need to find the total distance traveled by the object in that time interval and divide it by the total time.
First, let's integrate the given velocity function to find the displacement of the object. Since the object is moving along the x-axis, the displacement will give us the distance traveled.
∫(50 - 10t) dt = [50t - (10/2)t^2] from t=0 to t=6
Now, let's substitute the values:
[50(6) - (10/2)(6^2)] - [50(0) - (10/2)(0^2)]
= [300 - (10/2)36] - [0 - 0]
= [300 - 180] - [0 - 0]
= 120
So, the total distance traveled by the object from t=0 to t=6s is 120 units.
Now, let's calculate the average speed.
Average Speed = Total Distance / Total Time
Total Distance = 120 units
Total Time = 6s - 0s = 6s
Average Speed = 120 units / 6s = 20 units/s
Therefore, the average speed of the particle in the time interval t=0 to t=6s is 20 units/s.
To find the average speed of the particle, we need to calculate the total distance traveled and divide it by the total time taken.
Since velocity is given as v = (50 - 10t), we can integrate it with respect to time to find the displacement function. The displacement function, s(t), is the integral of the velocity function.
∫(50 - 10t) dt = 50t - 5t^2/2 + C,
where C is the constant of integration.
To find the displacement between t=0 and t=6s, we can evaluate the displacement function at these values:
s(6) - s(0) = (50*6 - 5*(6^2)/2 + C) - (50*0 - 5*(0^2)/2 + C)
= (300 - 90 + C) - (0 + 0 + C)
= 300 - 90.
Therefore, the displacement between t=0 and t=6s is 210 units.
Now, we divide the displacement by the total time taken to find the average speed:
Average speed = Total distance / Total time
= Displacement / Total time
= 210 units / 6 seconds
= 35 units/second.
Therefore, the average speed of the particle in the time interval t=0 to t=6s is equal to 35 units/second.
To find the average speed of the object in the given time interval, you first need to calculate the total distance traveled by the object during that time.
The average speed is given by the formula:
Average Speed = Total Distance / Total Time
In this case, the object is moving along the x-axis, so we can assume the distance traveled is the absolute value of the displacement.
To find the displacement during the time interval t = 0 to t = 6s, you can integrate the velocity function over that time interval.
Integrating the given velocity function: v = (50 - 10t)
∫(50 - 10t) dt = 50t - 5t^2/2
Evaluating the integral from t = 0 to t = 6:
(50 * 6 - 5 * 6^2/2) - (50 * 0 - 5 * 0^2/2) = 300 - 90 = 210 units
The displacement during the time interval t = 0 to t = 6s is 210 units.
Now, to calculate the average speed, you need to divide the total distance traveled (which is the absolute value of the displacement) by the total time interval (which is 6 seconds in this case):
Average Speed = 210 units / 6 seconds
Average Speed = 35 units per second
Therefore, the average speed of the particle in the time interval t = 0 to t = 6s is equal to 35 units per second.
recall that the average value of f(t) on the interval [a,b] is
∫[a,b] f(t) dt
-----------------
b-a
this works here because v is always positive in [0,6]