A minivan travels straight north in the right lane of a...

A minivan travels straight north in the right lane of a divided highway at $28.0 \mathrm{~m} / \mathrm{s}$. A camper passes the minivan and then changes from the left lane into the right lane. As it does so, the camper's path on the road is a straight displacement at $8.50^{\circ}$ east of north. To avoid cutting off the minivan, the north-south distance between the camper's back bumper and the minivan's front bumper should not decrease. (a) Can the camper be driven to satisfy this requirement? (b) Explain your answer.

Key Concepts

Relative Motion

Relative motion refers to analyzing the movement of one object from the perspective of another moving object. In problems involving vehicles or other moving bodies, we consider the velocities relative to a common frame or directly to one another to determine whether one object is approaching or receding from another in a specified direction.

Vector Decomposition

Vector decomposition is the process of breaking a vector into components along specified directions, such as north and east in navigation problems. This allows one to analyze how much of the velocity contributes to motion in each cardinal direction, which is critical when comparing movement along a particular axis.

Component Analysis

Component analysis involves examining each component of a vector separately to understand its effect on the overall motion. For example, by isolating the northward component of a velocity vector, one can compare it directly to another object's northward speed to determine whether their relative distance in that direction is increasing or decreasing.

Trigonometric Functions in Kinematics

Trigonometric functions, such as the cosine and sine, are used in kinematics to relate the angle of motion to the magnitude of its directional components. When a velocity vector is given at an angle, the cosine function provides the component along one axis (for example, northward), while the sine function gives the perpendicular component (for example, eastward).

Safety Constraints in Relative Motion

Safety constraints related to relative motion often require that a particular component of one object's velocity must not be lower than the corresponding component of another object's velocity. In merging or passing situations, ensuring that the relative separation in the critical direction (such as north–south) is maintained or increased is essential for avoiding collisions.

Transcript

00:01 Okay, question 9 .11.

00:05 A minivan's traveling north at 28 meters per second, the camper tries to pass and then change lanes.

00:14 Let's just kind of draw, we'll draw some vectors here.

00:16 So let's draw the minivan vector, and we're going to draw the minivan's velocity vector specifically.

00:27 So the m, and a camper passes the minivan, and then tries to change back into the.

00:37 Same lane so you have your camper over here passes the minivan and then tries to scoot back into that right lane and as it does so it's displacement so its velocity is at 8 .5 degrees east of north so basically have your camper velocity so it's going to be in this right -hand lane trying to cut back over into the left -hand lane right so to avoid cutting off the minivan, the distance between the two should not decrease.

01:26 Right, that makes sense.

01:28 So can the camper be driven to satisfy this requirement? okay, let's maybe draw another kind of overhead picture, so this makes a little more sense, hopefully.

01:42 So there's your minivan, here's your camper, some great overhead views here.

01:54 The minivan is moving directly north.

01:58 The camper is moving north as well, but it's trying to get into the right -hand lane, so it's moving slightly this way as well.

02:04 We have this distance.

02:05 Let's make the distance a little larger here.

02:07 We have this distance between the two.

02:14 Call it d.

02:16 We don't want that distance d to change.

02:19 That way the minivan doesn't get cut off.

02:21 So a, can you drive the camper to satisfy this requirement? yes.

02:30 Just have to drive fast enough, right? so as long as you drive fast enough in the camper, you won't cut the minivan off.

02:41 So let's figure out how fast you have to drive...