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 (iii) Normals to parametric curves or open all working together.

(iv)

Axial intercepts of a composite path

After TT seconds (T>2T>2) the particle reaches the point AA on the xx-axis. Find AA, and sketch the path, indicating the coordinates of any axial intercepts.

Explore this idea

Hint

  1. AA is where the normal meets y=0y=0.

  2. The curve itself crosses the xx-axis where t=e−2tt=e^{-2t}.

  3. Sketch the curve for 0≤t≤20\le t\le2 followed by the straight segment.

Particle path and enclosed region
Particle path and enclosed regionScroll across to see the full diagram.

Answer · (iv)

A=(4.15070373…,0)A=(4.15070373\ldots,0); initial point (1,−1)(1,-1) and curve x-intercept (0.852606…,0)(0.852606\ldots,0).

No y-axis intersection; follows curve until t=2, then its normal to A.

Explore the idea

Open interactive exploration

Curve until t = 2, then the normal to A

The fixed purple line is the normal. Jump to the key points.

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

The tangent follows the curve at the selected point. Its normal is perpendicular to it.

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

  • Path for 0 ≤ t ≤ 2
  • Normal from the collision to A
  • Tangent at the moving point
Key idea

The path has two pieces: a curve up to the collision, then a straight line to AA.

Ideas to try
Find where the curved part crosses the xx-axis.