Question

Two cars travel on the parallel lanes of a two-lane road. The cars are at the same location at time t=0s and move in such a way as to produce the velocity(relative to the ground) vs. time graph shown in the figure. On the graph, one vertical block is equivalent to one velocity unit.

Rank A through E from Largest to Smallest.

1. Rank car #1’s speed relative to the ground at the lettered times (A through E).

2. Rank car #1’s speed relative to car #2 at the lettered tiles (A through E).

3. Rank the distance between the cars at the lettered tiles.

Answer 1

The concept used to solve this question is velocity.

Frist use the fact that the speed is the magnitude of the velocity, but the graph is expressed for velocity. Then using the magnitude of the velocity rank the speed relative to the ground.

Secondly, the speed of car 1 relative to car 2 can be calculated using by assuming the speed of car 1 relative to a fixed point and the velocity of car 2 to the same fixed point. Then the magnitude of the difference in velocities gives the speed of one car relative to the other.

Finally, the distance between the cars is calculated using the fact that the cars begin at the same location, It means their initial separation is zero, and the distance increases gradually from point A to point E.

Fundamentals

The speed is equal to the magnitude of the velocity.

${\\rm{S}} = \\left| {\\vec v} \\right|$

Here, S is the speed and $\\vec v$ is the velocity vector.

(a)

Rank car 1's speed relative to the ground.

Refer to the graph of velocity versus time of car #1.

The magnitude of the speed is highest at point E.

The magnitude of speed at A is smaller than E.

The magnitude of the speed at point C is zero and smallest of all.

The magnitude of the speed at point D is smaller than E and A.

The magnitude of the speed at point B is smaller than E, A, D, and larger than C.

Part a

The rank of car 1's speed relative to the ground from largest to smaller is ${\\rm{E}} > {\\rm{A}} > {\\rm{D}} > {\\rm{B}} > {\\rm{C}}$.

The magnitude of the speed is shown at various points is shown in the picture below:

The graph clearly shows the magnitude of the velocities at various points in red arrows.

Always look for the magnitude of the velocity, not the velocity only.

The speed of car 1 relative to car 2 is equal to the magnitude of the difference in velocities.

(b)

Rank car 1'speed relative to car 2 as follows:

Refer to the graph of velocity versus time.

At point A car 1 is in the negative direction and car 2 is in a positive direction.

At point D car 1 is in the positive direction and the car is in the positive direction

The speed of car 1's relative to car 2's is highest at point A and lowest at point E.

The speed of car 1's relative to car 2's is smaller at point B than A and larger than C, D, and E.

Part b

The rank of speed of car 1's relative to car 2\u2019s is ${\\rm{A}} > {\\rm{B}} > {\\rm{C}} > {\\rm{D}} > {\\rm{E}}$.

The magnitude of car 1's speed relative to car 2's shown in the picture below:

The arrows clearly show the speed of car 1's relative to the speed of car 2's.

(c)

Rank the distance between the cars at various points.

At time A, the two cars begin at the same location, so their initial separation is zero.

From point A to point C, the cars move in opposite directions.

From point C, they began to travel in the same direction, since they start at the same point, the distance between them initially is zero. Finally, the distance gradually increases from point A to point E.

Part c

The rank of the distance between the cars at various point is ${\\rm{E}} > {\\rm{D}} > {\\rm{C}} > {\\rm{B}} > {\\rm{A}}$ .

The below picture shows the velocity versus time graph for car 1 and car 2.

The initial separation is zero for both cars.