Understanding Acceleration: Calculating Distance Covered in 3 Seconds

[sangfroid]

Hi,

Lets say an object is initially at rest and then accelerates at a rate of 30m/s2. This means that every second, its velocity will get increased by 30/ms. By the 1st second, its velocity will be 30m/s. By the 2nd second, its velocity will be 60m/s and by the 3rd second, its velocity will be 90m/s.

So in the 1st second, it will cover distance of 30m. In the 2nd second, it will cover distance of 60m and in the 3rd second, it will cover distance of 90 m. So by 3rd second, it will be covering a total distance of 180m.

I assume so far I am correct.

But say, if I wish to calculate distance covered in 3 seconds using "s=ut+1/2 at2" formula, I get the distance covered as = 1/2 * 30 (sq 3) = 135 meters.

Where am I missing something wrong ? What is going wrong over here ??

 

[Superstring]

Hi,

Lets say an object is initially at rest and then accelerates at a rate of 30m/s2. This means that every second, its velocity will get increased by 30/ms. By the 1st second, its velocity will be 30m/s. By the 2nd second, its velocity will be 60m/s and by the 3rd second, its velocity will be 90m/s.

So in the 1st second, it will cover distance of 30m. In the 2nd second, it will cover distance of 60m and in the 3rd second, it will cover distance of 90 m. So by 3rd second, it will be covering a total distance of 180m.

I assume so far I am correct.

But say, if I wish to calculate distance covered in 3 seconds using "s=ut+1/2 at2" formula, I get the distance covered as = 1/2 * 30 (sq 3) = 135 meters.

Where am I missing something wrong ? What is going wrong over here ??

You are forgetting that the velocity doesn't just suddenly become 30 m/s, then 60, then 90. It hits every velocity between them between each second, so the x=vt equation that you used to get the 30, 60, and 90 m is incorrect because the velocity is changing with time.

The best way to show this is graphically:

With the way you interpreted the problem, a graph of the velocity over time would look something like this (ignore scale, y intercept, etc.):

However it should really look like this (ignore scale, y intercept, etc.):

 

[sangfroid]

couldn't still understand it clearly...

 

[Superstring]

couldn't still understand it clearly...

Your object starts from rest and has an acceleration of 30 m/s². This means that when t=1 second, the object will be traveling at 30 m/s. It doesn't stay at this speed for the entire second and then magically reach 60 m/s at t=2; at t=1.5, the object is traveling at 45 m/s. At t=1.1, it's traveling at 33 m/s. You can choose any point in between t=1 and t=2 and the speed will NOT be just 30 m/s. The s=vt equation simply doesn't work when there's an acceleration involved. Between t=1 and t=2, it doesn't travel 30 m because it wasn't traveling at the same speed the entire time.Here's where the s = v₀t + 1/2 at² equation came from:

If you have a graph of velocity vs. time, the area under the function between two points in time represents the change in displacement between those two points in time.

If you have something moving at constant velocity, the velocity function is a horizontal line. This means that the area under the function between two points in time is a rectangle, with area = the base (change in time) times the height (velocity):

http://openlearn.open.ac.uk/file.php/3268/!via/oucontent/course/434/s207_2_017i.jpgIn uniformly accelerated motion, the velocity function is a line with the initial velocity as the y-intercept and the acceleration as the slope. The change in displacement is still the area under the function, but this can no longer be calculate by multiplying the velocity by the change in time, because the shape is no longer a rectangle. The area can be broken up into two shapes: a rectangle and a triangle. Calculating this gives the equation s = v₀t + 1/2 at².

http://openlearn.open.ac.uk/file.php/3268/!via/oucontent/course/434/s207_2_035i.jpg

 

[Lsos]

So in the 1st second, it will cover distance of 30m.

IF it was going 30m/s for the entire second. It was NOT. It started out not moving at all (0m/s), and only at the end of the second it was going 30m/s.

It's average speed over the entire second was only 15m/s.