Question

In a historical movie, two knights on horseback start from rest 86 m apart and ride directly toward each other to do battle. Sir George's acceleration has a magnitude of 0.21 m/s2, while Sir Alfred's has a magnitude of 0.26 m/s2. Relative to Sir George's starting point, where do the knights collide?

Answer 1

Relative to Sir George's starting point, the knights collide at a distance of 38.43 m from Sir George's starting point.

Step-by-step explanation:

Let the distance covered by Sir George be
`S_(1)`

and the distance covered by Sir Alfred be
`S_(2)`

Since the knights collide, hence they must have traveled for the same amount of time just before collision

From one of the equations of motion for linear motion

`S = ut + (1)/(2)at^(2)`

Where
`S` is the distance traveled

`u` is the initial velocity

`a` is the acceleration

and
`t` is the time

For Sir George,

`S = S_(1)`

`u` = 0 m/s (Since they start from rest)

`a =`0.21 m/s²

Hence,

`S = ut + (1)/(2)at^(2)` becomes

`S_(1) = (0)t + (1)/(2)(0.21)t^(2)\\S_(1) = 0.105 t^(2)\\`

`t^(2) = (S_(1))/(0.105)`

Now, for Sir Alfred

`S = S_(2)`

`u` = 0 m/s (Since they start from rest)

`a =`0.26 m/s²

Hence,

`S = ut + (1)/(2)at^(2)` becomes

`S_(2) = (0)t + (1)/(2)(0.26)t^(2)\\S_(2) = 0.13 t^(2)\\`

`t^(2) = (S_(2))/(0.13)`

Since, they traveled for the same time,
`t` just before collision, we can write

`(S_(1))/(0.105)= (S_(2))/(0.13)`

Since, the two nights are 86 m apart, that is, the sum of the distances covered by the knights just before collision is 86 m. Then we can write that

`S_(1) + S_(2) = 86` m

Then,
`S_(2) = 86 - S_(1)`

Then,

`(S_(1))/(0.105)= (S_(2))/(0.13)` becomes

`(S_(1))/(0.105)= (86 -S_(1))/(0.13)`

`0.13{S_(1)}= 0.105({86 -S_(1)})\\0.13{S_(1)}= 9.03 - 0.105S_(1)}\\0.13{S_(1)} + 0.105S_(1)}= 9.03 \\0.235{S_(1)} = 9.03\\{S_(1)} =(9.03)/(0.235)`

`S_(1) = 38.43` m

∴ Sir George covered a distance of 38.43 m just before collision.

Hence, relative to Sir George's starting point, the knights collide at a distance of 38.43 m from Sir George's starting point.