Answer:
(C) t = 8 sec
(B) xy = 160 m
Explanation:
To find the time (t) it takes to cover the distance (xy), begin by creating a displacement equation for each cyclist using the following SUVAT equation:
[tex]s = ut + \dfrac{1}{2}at^2[/tex]
where:
- s = displacement (m)
- u = initial velocity (m/s)
- a = acceleration (m/s²)
- t = time (s)
In this scenario, take the cyclists' initial velocities (u) to be their speeds when they cross point x.
Cyclist 1
- s = xy
- u = 16 m/s
- a = 1 m/s²
Substitute the values into the SUVAT equation:
[tex]xy = 16t + \dfrac{1}{2}(1)t^2\\\\\\xy = 16t + \dfrac{1}{2}t^2[/tex]
Cyclist 2
- s = xy
- u = 12 m/s
- a = 2 m/s²
Substitute the values into the SUVAT equation:
[tex]xy = 12t + \dfrac{1}{2}(2)t^2\\\\\\xy = 12t + t^2[/tex]
Now, set the equations for xy equal to each other and solve for t:
[tex]12t+t^2=16t+\dfrac{1}{2}t^2\\\\\\\dfrac{1}{2}t^2-4t=0\\\\\\t^2-8t=0\\\\\\t(t-8)=0\\\\\\t=0,t=8[/tex]
So, the two cyclists are at the same point in the race at t = 0 (when they are at point x) and at t = 8 (when they are at point y). Therefore, the time required to cover the distance xy is:
[tex]\Large\boxed{\boxed{t=8\; \sf sec}}[/tex]
(Note that t = 0 marks the beginning of this particular leg of the race between points x and y rather than the beginning of the entire race).
To find the distance xy, we can substitute t = 8 into one of the equations for xy. Let's use xy = 12t + t²:
[tex]xy = 12(8) + (8)^2\\\\xy = 96 + 64\\\\xy = 160[/tex]
Therefore, the distance xy is:
[tex]\Large\boxed{\boxed{xy=160\; \sf m}}[/tex]
