Question

A car is traveling due west at 20.0 m/s. Find the velocity of the car after 37.00 s if its acceleration is 1.0 m/s2 due east. Assume that the acceleration remains constant. a) 17.0 m/s west b) 17.0 m/s east c) 23.0 m/s west d) 23.0 m/s east e) 11.0 m/s south

Short answer

a) 20.0 m/s east b) 17.0 m/s east c) 20.0 m/s west d) 17.0 m/s west Answer: b) 17.0 m/s east

Step-by-step solution

Write down the known information

Initial velocity (v₀) = 20.0 m/s (west) Acceleration (a) = -1.0 m/s² (east) Time (t) = 37.00 s

Use the formula for velocity with constant acceleration

v = v₀ + a*t

Plug in the values and solve for the final velocity

v = 20.0 m/s + (-1.0 m/s² * 37.00 s) v = 20.0 m/s - 37.0 m/s v = -17.0 m/s Since the value we got for v is negative, it means that the velocity is directed to the east, making the final answer: v = 17.0 m/s (east)

Match the answer with the available options

The final velocity of the car is 17.0 m/s east, which corresponds to option b).

Key concepts

Initial Velocity

When discussing motion, the term initial velocity refers to the velocity of an object at the start of a period of time. It's an essential part of predicting an object's future position and velocity, especially when the object is under controlled conditions, such as a car accelerating on a road.

Imagine a car traveling at a steady speed. The speed of the car when you first check the speedometer is the initial velocity. In the given exercise, the car's initial velocity is 20.0 m/s directed due west. Understanding the initial velocity is crucial because, combined with acceleration, it helps us foresee how the car's speed and direction might change over time.

Constant Acceleration

In physics, constant acceleration indicates that an object's speed increases or decreases by the same amount each second. The key point is the 'constant' part—it means that the change in the object's motion is predictable and steady over time, simplifying our calculations drastically.

In our scenario, the car has a constant acceleration of 1.0 m/s² in the direction due east. Even though we often associate acceleration with speeding up, it technically means any change in velocity, which includes slowing down or changing direction. Here's a critical note: when acceleration is in the opposite direction of the velocity (like in our exercise), it will decrease the object's speed.

Kinematic Equations

To analyze motion, we use kinematic equations, which connect velocity, acceleration, and displacement during a time interval. They are the bread and butter of solving motion problems in physics when objects move with a uniform acceleration.

The kinematic equation for velocity, which is of particular interest here, is given by the formula: \( v = v_0 + at \). In this formula, \( v \) is the final velocity, \( v_0 \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time.

This equation tells us how the initial velocity and acceleration will combine to give a new velocity after a certain amount of time. It's a mathematical translation of how an object's motion evolves.

Direction of Velocity

Velocity not only tells us about the speed of an object but also the direction of its motion. It's a vector quantity, which means direction is as important as magnitude. This is why we describe the car's initial velocity heading west and the acceleration towards the east.

In calculations, direction impacts the sign of our values. For instance, if we give west a positive sign, east would get a negative sign. This connection of velocity and direction plays a significant role in understanding the behavior of moving objects and provides a complete description of their motion.

For our car, a negative sign in the calculated final velocity indicated that the direction of the car's velocity has shifted to the direction of acceleration, which is east in this case. It's an essential concept for interpreting the results correctly and choosing the right answer among multiple choices.