At what time do the two cars collide?

Question

Two stunt drivers drive directly toward each other. At time t=0 the two cars are a distance D apart, car 1 is at rest, and car 2 is moving to the left with speed v. Car 1 begins to move at t=0, speeding up with a constant acceleration a. Car 2 continues to move with a constant velocity.

1。At what time do the two cars collide?

2。Find the speed of car 1 just before it collides with car 2.

Answer 1

1. Distance = D

Intial relative speed = v

Relative acceleration = a

Time t = ?

D = vt + 1/2 at^2

Or 1/2 at^2 + vt - D = 0

Or t^2 + (2v/a)t - 2D/a = 0

Or t = [-2v/a + sqrt{(2v/a)^2 - 4*1(-2D/a)}]/2

Or t = [-2v/a + sqrt(4v^2/a^2 + 8D/a)]/2

Or t = [-2v/a + sqrt{4(v^2/a^2 + 2D/a)}]/2

Or t = [-2v/a + 2*sqrt(v^2/a^2 + 2D/a)]/2

Or t = -v/a + sqrt(v^2/a^2 + 2D/a)

Or t = -v/a + sqrt{v^2/a^2(1 + 2Da/v^2)}

Or t = -v/a + v/a sqrt(1 + 2Da/v^2)

Or t = v/a sqrt(1 + 2Da/v^2) - v/a

Or t = (v/a)[sqrt(1 + 2Da/v^2) - 1]

2. Speed of car 1 at time t = 0 + at

= at

= v*[sqrt(1 + 2Da/v^2) - 1]

Ans:

1) (v/a)[sqrt(1 + 2Da/v^2) - 1]

2) v*[sqrt(1 + 2Da/v^2) - 1]

Answer 2

Hi

1. First, let S1 and S2 be the distance moved by car 1 and car 2 respectively (from their positions at t=0 to where they collide)

∴ S1 + S2 = D

let t be the time after t=0 when the 2 cars collide. So..

S2 = vt

S1 = ut + 0.5at^2 where u, initial velocity of car 1 = 0

∴  vt + 0.5at² = D

and t = {-v ± √[v²-4(0.5a)(-D)]} / a = -v/a ± √(v²+2aD)/a

2. v² = u² + 2aS1 = 2aS1 since u = 0

∴ v = √[2a(D-S2)] = √[2a(D-vt)]

Hope this helps=)