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Calculus, EXT1 C3 EQ-Bank 11

The maximum speed that a hammer can reach after falling vertically from the top of a skyscraper is `u\ text{ms}^(-1)`.

The hammer's  speed, `v\ text{ms}^(-1)`, after falling `x` metres, is given by the differential equation 

`(dv)/(dx)=c(u^2-v^2)/v`  where `c` is a positive constant.

Find an expression of `v` in terms of `x`.    (4 marks)

Show Answers Only

`v=u(1-e^(-2cx))^(1/2)`

Show Worked Solution
`(dv)/(dx)` `=c((u^2-v^2)/v)`  
`v/(u^2-v^2)\ dv` `=c\ dx`  
`int v/(u^2-v^2)\ dv` `=int c\ dx`  
`-1/2ln(u^2-v^2)` `=cx+C`  

 

`text{When}\ \ x=0, v=0:`

`-1/2ln\ u^2` `=c xx 0+C`  
`C` `=-ln\ u`  
`-1/2ln(u^2-v^2)` `=cx-ln\ u`  
`ln(u^2-v^2)` `=-2cx+2ln\ u`  
`ln(u^2-v^2)` `=-2cx+ln\ u^2`  
`u^2-v^2` `=e^(-2cx+ln\ u^2)`  
`u^2-v^2` `=e^(-2cx)*e^(ln\ u^2)`  
`u^2-v^2` `=u^2e^(-2cx)`  
`v^2` `=u^2-u^2e^(-2cx)`  
  `=u^2(1-e^(-2cx))`  
`v` `=+-u(1-e^(-2cx))^(1/2)`  
  `=u(1-e^(-2cx))^(1/2),\ \ (v>0)`  

Calculus, EXT1 C3 2021 SPEC1 7

The velocity of a particle satisfies the differential equation  `(dx)/(dt) = xsin(t)`,  where  `x`  centimetres is its displacement relative to a fixed point `O` at time `t` seconds.

Initially, the displacement of the particle is 1 cm.

  1. Find an expression for `x` in terms of `t`.  (3 marks)

  2. Find the maximum displacement of the particle and the times at which this occurs.  (2 marks)

Show Answers Only
  1. `x = e^(1 – cos(t))`
  2. `x_text(max) = e^2\ \ text(when)\ \ t = pi, 3pi, 5pi, …`

     

    `text(or)\ t = (2k + 1)pi\ \ text(for integral)\ \ k =0,1,2,…`

Show Worked Solution
a.    `(dx)/(dt)` `= x sin(t)`
  `int 1/x\ dx` `= int sin(t)\ dt`
  `log_e x` `= -cos(t) + c`

 

`text(When)\ \ t = 0, x = 1`

`log_e 1` `= -cos0 + c`
`c` `= 1`
`log_e x` `= -cos(t) + 1`
`:. x` `= e^(1 – cos(t))`

 

b.    `x` `= e^(1 – cos(t))`
  `(dx)/(dt)` `= sin(t) · e^(1 – cos(t))`

`text(Find)\ \ t\ \ text(when)\ \ (dx)/(dt) = 0:`

`e^(1 – cos(t)) != 0`

`sin(t) = 0\ \ text(when)\ \ t = 0, pi, 2pi, …`

`x_text(max) = e^2\ \ text(when)\ \ t = pi, 3pi, 5pi, …`

`text(or)\ \ t = (2k + 1)pi\ \ text(for integral)\ \ k =0,1,2,…`