Differentiation + integration: how to solve for acceleration and displacement given a specific velocity time graph

calculusderivativesintegrationproblem solvingsolution-verification

The velocity-time graph shown below is for a particle moving in a straight line, from rest at A, through B to C and then back to rest at B.

I have a few questions below regarding this velocity-time graph that I am unsure how to solve, any hints would be much appreciated!

enter image description here

My attempts so far:

What is the acceleration of the particle 13 seconds after leaving A?

I know that differentiating velocity with respect to time gives acceleration – however I am unsure of how to apply this in the context of a graph. I'm also thinking I can apply this formula that I found where:

acceleration ($m/s^2$) = change in velocity ($m/s$) ÷ time taken ($s$)

Using the formula I got: $(10+7) ÷ 13 = 1.3077$ – but this is wrong. The correct answer is $-2.5m/s^2$ – I don't know how to get there though.

What is the particle's displacement from A 28 seconds after leaving A?

I think I have confused this with distance in the next question – the answer for displacement given is $85$m but how do you get there?

How far does the particle move in the first 28 seconds?

Distance is given from area under the given curve (from $0$ to $28$ seconds)

So I've added the area like so:

$(5 \times 10)/2 = 25$

$7 \times 10 = 70$

$(4 \times 10)/2 = 20$

$(12 \times 5)/2 = 30$

$25 + 70 + 20 + 30 = 145$m

This is from a Year 12 Methods textbook.

Thanks very much in advance for any tips!

Best Answer

First Question:

acceleration (m/s2) = change in velocity (m/s) ÷ time taken (s)

here calculate a for 12 to 16 seconds.

for that

a=dv/dt

=0-10/16-12
=-2.5 m/s2

Second Question:

In v/t graph you can find displacement by calculating the area of the graph. Because displacement equal to velocity into total taken time.

we can divide this graph into four part:

a1 = 0 to 5 s

a2 = 5 to 12 s

a3 = 12 to 16 s

a4 = 16 to 28 s

a1 = 5*10/2
   =25m

a2 = 7*10
   = 70m

a3 = 4*10/2
   = 20m

a4 = 12*(-5)/2
   = -30m

Total displacement = Total aria
                   = a1 + a2 + a3 + a4
                   = 25 + 70 + 20 -30
                   = 85m
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