A particle switches from a curve to its normal

A particle is moving along a curve, CC, such that its position at time tt seconds after it is set into motion is given by the parametric equations x=t+e−2tx=t+e^{-2t}, y=t−e−2ty=t-e^{-2t}.

(i) State the coordinates of the initial position of the article. mark

(ii) Explain what would happen to the path of the particle after a long time. mark

At the time of 2 seconds after the particle was set into motion, an external force struck the particle resulting in the particle moving in a straight line along the normal to the path at the point of collision.

(iii) Find an equation for the normal to the curve CC at the point t=2t=2, leaving your answer correct to 3 decimal places. marks

After TT seconds, where T>2T>2, the particle reaches point AA, which lies on the xx-axis, and stops moving.

(iv) Find the coordinates of the point AA. Hence, give a sketch of the path traced by the particle, indicating the coordinates of any axial intercepts. marks

(v) Find the total area bounded by the path of the particle in the first TT seconds and the positive xx-axis. marks

[2017/SRJC/Prelim/I/10]

Total marks: 13

Solution

If this part uses an earlier result, revisit (i) Parametric curves or open all working together.

(ii)

Long-term behaviour

Explain what would happen to the path of the particle after a long time (with no impact).

Explore this idea

Hint

  1. Let t→∞t\to\infty and see which terms survive.

Answer · (ii)

Original parametric path approaches y=xy=x as t→∞t\to\infty.

Explore the idea

Open interactive exploration

The path hugs y = x as time goes on

Slide towards the right end: the tangent becomes almost parallel to y = x (gradient near 1).

Position along the curve(1, -1)t = 0
Try an example
Tangent gradient
-3

Start (1,−1)(1,-1): the gradient is 1+21−2=−3\frac{1+2}{1-2}=-3, and the particle begins by heading up and to the left before the curve turns.

For finite nonzero gradients, tangent gradient × normal gradient = −1. Both axes use equal scales.

  • Path
  • Y = x
  • Tangent
Key idea

For large tt, dydx→1\frac{dy}{dx}\to1 and the curve is asymptotic to y=xy=x.

Ideas to try
Slide to the far right: what is the gradient?