Unit 6 Test Part B MCR3U Summer
4. Sophie is riding her bike home when she runs over a nail. It gets stuck to the tire of her bike but does not pop
the tire. As she continues to cycle home, the nail hits the ground every 2 seconds and reaches a maximum
height of 48 cm

You did not ask anything, i guess that you want to find the equation of the height of the nail as a function of time.

Data that we have:

The nail is in a circular object.

the nail hits the ground every 2 seconds, then the weel does a full cycle every 2 seconds.

The maximum height of the nail is 48cm, and as the nail is stuck on the weel, the maximum height will be when the nail is on top of the weel.

From this, we can conclude that the diameter of the weel is 48cm.

Now, with this we could write the equation of the height of the nail as a function of time.

We know that this is a periodic function, so let's write it as:

f(t) = A*cos(c*t) + B

where A, B and c are constanst.

At t = 0, when the byke hits the nail, the nail must be in the ground, so we have that:

0 = A*cos(c*0) + B

A + B = 0

A = - B

and we also know that the function is periodic with T = 2s

cos(c*t) = cos(c*(t + 2s) = cos( c*t + 2s*c)

and we know that the period of the cosine function is 2*pi

then:

c*2s = 2*pi

c = (2pi/2s) = pi/s

then our function is:

f(t) = -B*cos(t*pi/s) + B

now, the maximum heightwill be when cos(t*pi/s) = -1, and this happens at t = 1s

f(1s) = -B*cos(pi) + B = 2B

And we know that the maximum height is 48cm

2*B = 48cm

B = 24cm

Then the function of the height of the nail is:

f(t) = -24cm*cos(t*pi/s) + 24cm

other answer is correct. i had used the radius of the radius somehow. a website: