Question

The head of a rattlesnake can accelerate \(50 \mathrm{~m} / \mathrm{s}^{2}\) in striking a victim. If a car could do as well, how long would it take for it to reach a speed of \(100 \mathrm{~km} / \mathrm{h}\) from rest?

Short answer

It would take the car 0.56 seconds to reach a speed of 100 km/h from rest if it could accelerate at 50 m/s².

Step-by-step solution

Convert Velocity to Compatible Units

First convert the final velocity from kilometers per hour to meters per second. 100 km/h is equal to approximately 27.78 m/s.

Apply the Acceleration Formula

Using the formula for acceleration (v = u + at), where v = 27.78 m/s, u = 0 m/s and a = 50 m/s², solve for t. Substituting the values into the formula, we get 27.78 = 0 + 50t

Calculate Time

Rearranging the acceleration formula, we get the time t = 27.78 / 50 = 0.56 seconds.

Key concepts

Kinematics

Kinematics is the branch of classical mechanics that describes the motion of points, bodies, and systems of bodies without considering the forces that cause them to move. At the core of kinematics is the study of an object's velocity and acceleration. Velocity is the speed of an object in a particular direction, while acceleration is the rate at which an object's velocity changes over time.

When studying kinematics, we often look at various graphs and equations to model motion. For instance, velocity-time graphs can help us understand how an object's speed changes and what that implies about the object's acceleration. An important aspect of kinematics is the distinction between uniform motion (at constant velocity) and uniformly accelerated motion (at constant acceleration). In your rattlesnake example, the snake's head is described as having a constant acceleration, which is a classic case of uniformly accelerated motion.

By using kinematic equations, we can predict future states of motion, such as where an object will be after a certain period and how fast it will be going.

Unit Conversion

In physics, unit conversion is a critical step that students must master to solve problems accurately. Different systems of measurement can be used depending on the context, such as the International System of Units (SI) for scientific measurements. The SI unit for measuring velocity or speed is meters per second (m/s), while for everyday contexts, kilometers per hour (km/h) is commonly used.

To convert units, you typically use a conversion factor that represents the ratio of one unit to another. For example, to convert from km/h to m/s, you can use the conversion factor where 1 km/h equals approximately 0.2778 m/s. This is what was done in Step 1 of the solution, where the car's speed was converted into meters per second before using the acceleration formula.

Understanding and applying unit conversion is essential because it ensures that all the units in a formula are consistent, preventing errors in calculation. A failure to convert units correctly can lead to incorrect results and a misinterpretation of the motion being analyzed.

Motion Equations

Motion equations, also known as the equations of motion, are mathematical relationships that describe an object's kinematics. These equations enable us to predict the final position, velocity, and time elapsed for an object under uniform acceleration. To solve the rattlesnake head acceleration problem, the equation used was one of the basic motion equations given by: \( v = u + at \), where \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time.

In problems involving constant acceleration, the final velocity can be found if you know the initial velocity, acceleration, and time. Conversely, if you know the final and initial velocities, and the acceleration, you can find the time as shown in Step 3. There are several motion equations, and they are typically taught as part of introductory physics courses because they apply to a wide range of everyday phenomena.

These equations underpin much of classical mechanics and are vital tools in engineering, physics, and various applied sciences. They are simple yet powerful, allowing prediction of an object’s future motion from the current state of motion and known forces.