Question

Speedy Sue, driving at 34.0 m/s, enters a one-lane tunnel. She then observes a slow-moving van 175 m ahead traveling at 5.50 m/s. Sue applies her brakes but can accelerate only at −1.00 m/s² because the road is wet. Will there be a collision?

Yes

If yes, determine how far into the tunnel and at what time the collision occurs. If no, determine the distance of closest approach between Sue's car and the van. (If no, enter "0" for the time.)

distance	m
time	s

Answer 1

Solution

Speed of Speedy Sue with which she enters the tunnel, u = 34.0 m/s

Initial distance between the van and Speedy Sue = 175 m

Speed with which the van travels, v_van = 5.50 m/s

Acceleration of Sue = -1.00 m/s²

Assumption: The van and Speedy Sue are travelling in same direction inside the tunnel.

The collision will only occur if Sue and van are at same position inside the tunnel at same time. As the road is wet maximum retardation possible for Sue is 1 m/s².

Let the collision occurs at a distance x from the van's initial position after time t.

Distance travelled by van in time t is given by

(1)

Distance travelled by Speedy Sue after time t is

(2)

Substitute equation (1) in equation (2)

or

As we are getting real values of t means the assumption that collision will occur is true.

Sue collides with the van after 7 s of travel inside the tunnel.

Distance covered by Speedy Sue before the collision is given by

As the collision occurs at t = 7 s so ignore t = 50 s for any calculation.

So, the collision occurs at a distance of 213.5 m inside the tunnel after 7s of travel.

Yes, there will be a collision.