Problem 6 - Calculus - AP Edition

Question

An automobile, initially at rest, begins to move along a straight track. The velocity increases steadily until suddenly the driver sees a concrete barrier in the road and applies the brakes sharply at time \(t_{0} .\) The car decelerates rapidly, but it is too late - the car crashes into the barrier at time \(t_{1}\) and instantaneously comes to rest. Sketch a position versus time curve that might represent the motion of the car. Indicate how characteristics of your curve correspond to the events of this scenario.

Step-by-Step Solution

Answer

The curve starts concave up, becomes concave down after braking, and levels off at the crash.

Step 1: Understanding the Motion

The car starts at rest, so the initial velocity and position are both zero, i.e., \(v(0) = 0\) and \(x(0) = 0\). As it moves along the track, the car accelerates uniformly until the driver applies the brakes at time \(t_0\). From then on, it decelerates until it comes to a complete stop at time \(t_1\).

Step 2: Sketching the Graph

On a position vs. time graph, the car starting from rest means it starts at the origin (0,0). The car accelerating uniformly is shown as a curve that gets steeper over time, indicating increasing velocity. At time \(t_0\), the driver hits the brakes sharply causing the curve to start flattening since velocity is decreasing until \(t_1\), when the car stops, and the curve should level off horizontally, indicating no further changes in position.

Step 3: Characteristics of the Curve

Initially, the position vs. time graph is a gradually steepening (concave up) curve, representing constant acceleration. At \(t_0\), the curve changes from a steep concave to a convex shape due to sharp deceleration. At \(t_1\), the curve becomes a straight horizontal line, indicating the car has stopped moving as it hits the barrier.

Key Concepts

Understanding Uniform Acceleration

When we talk about uniform acceleration, we mean that the object's velocity increases by the same amount in each equal time interval. This occurs when a constant force acts on a body. In the context of a car, it means pressing the pedal consistently to increase speed.
For example, if a car's speed increases from 0 to 60 km/h in 10 seconds, and then to 120 km/h in another 10 seconds, this is uniform acceleration. Mathematically, we can express acceleration as: \[ a = \frac{\Delta v}{\Delta t} \]where \(\Delta v\) is the change in velocity, and \(\Delta t\) is the change in time.
On the position vs. time graph, this uniform acceleration is shown by a curve that gets steadily steeper. This is because the car's velocity is increasing, and thus it covers more distance in a shorter time.
A curve that steepens upwards (concave up) signifies that velocity is not just increasing but doing so at a constant rate.

Deceleration and its Effects

Deceleration is essentially negative acceleration—this means a reduction in the speed of an object. In the scenario of a car approaching a barrier, deceleration kicks in when the driver applies the brakes. This sudden reduction in velocity is critical in stopping the vehicle.
Just as uniform acceleration results in a naturally curved position vs. time graph, so does deceleration, but in the opposite manner. When the brakes are applied, the steeply rising curve you see during acceleration starts to bend outwards or flatten, transitioning into a convex shape. This indicates that the car's speed is reducing.
The position graph provides a visual tale of what's happening during deceleration:

• The curvature starts to decrease, signifying a drop in slope (velocity).
• Eventually, just before the car stops, the curve levels off to a horizontal line, marking zero velocity.

Adjusting speed by deceleration involves gradual pressure until the speed desired, or in this case, until a complete stop, denoted by the car crashing into the barrier.

Interpreting a Position vs. Time Graph

A position vs. time graph is a powerful tool to visualize motion. When analyzing such a graph, it's crucial to know how to interpret the slopes and curvatures, as they provide insight into the dynamics of motion.
Initially, for the car at rest, the graph starts at the origin \((0,0)\). As the car accelerates uniformly, the graph gradually steepens. This steepening indicates an increase in velocity.
When the brakes are applied, causing deceleration, the curve begins to flatten. This change from steep slope to a flatter one signals decreasing velocity. Eventually, as the car halts upon impact with the barrier, the graph transitions to a horizontal line, indicating a stop in motion.
Interpreting these changes require an understanding that:

• The slope of the graph represents the velocity of the car.
• The concavity (curving upwards or downwards) indicates acceleration or deceleration.
• A horizontal segment denotes a complete stop.

Recognizing these attributes helps in accurately depicting and predicting the car's motion throughout the journey.