Question

The red kangaroo (Macropus rufus, shown in Figure 2.5) is the largest marsupial in the world. It has been clocked hopping at a speed of \(65 \mathrm{~km} / \mathrm{h}\). (a) How far (in kilometers) can a red kangaroo hop in \(3.2\) minutes at this speed? (b) How much time will it take the kangaroo to hop \(0.25 \mathrm{~km}\) at this speed?

Short answer

(a) The kangaroo can hop 3.4645 km in 3.2 minutes. (b) It takes about 0.23 minutes to hop 0.25 km.

Step-by-step solution

Convert Time from Minutes to Hours

To find out how far the kangaroo can hop in a given time at a certain speed, first, we need to express the time in hours because the speed is given in kilometers per hour.3.2 minutes = \( \frac{3.2}{60} \) hours.Calculating this gives:3.2 minutes = \( \frac{3.2}{60} = 0.0533 \) hours (rounded to four decimal places).

Calculate the Distance

We know the speed of the kangaroo is 65 km/h, and we just calculated the time.To find the distance the kangaroo can hop, use the formula:\[\text{Distance} = \text{Speed} \times \text{Time} = 65 \text{ km/h} \times 0.0533 \text{ hours}\]Calculating this gives:Distance \( = 3.4645 \) km (rounded to four decimal places).

Set Up the Equation for Time

For the second part, we need to find out how much time it will take to hop 0.25 km at a speed of 65 km/h. The formula is similar, but this time, we solve for time.\[\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{0.25 \text{ km}}{65 \text{ km/h}}\]Calculating this gives:Time \( = 0.003846 \) hours (rounded to six decimal places).

Convert Time from Hours to Minutes

Since the time is typically more useful in minutes for short durations, convert the time from hours to minutes:\[0.003846 \text{ hours} \times 60 \text{ minutes/hour} = 0.23076 \text{ minutes}\]Rounded to five decimal places, it becomes approximately 0.23 minutes.

Key concepts

Speed and Velocity

When we talk about speed in kinematics, we are referring to how fast an object is moving, regardless of its direction. This is what we call a scalar quantity. In contrast, velocity is a vector quantity because it addresses both speed and direction. For example, hopping at a speed of 65 km/h might sound impressive, but without knowing the direction, we can't call it a velocity.

Speed is generally more straightforward to calculate and involves simple formulas that usually multiply the rate (speed) by time to find distance or divide distance by rate to find time. This detail is pivotal when solving problems where the direction is not relevant, like how far a kangaroo can hop in a given time.

Unit Conversions

Unit conversions are essential in solving kinematics problems accurately, especially when the units you're working with don't match. For instance, in our kangaroo problem, speed is given in kilometers per hour, but time is in minutes. That's an automatic cue for a conversion!

To convert minutes to hours, divide the number of minutes by 60, since there are 60 minutes in an hour. Therefore, 3.2 minutes becomes \( \frac{3.2}{60} = 0.0533 \) hours. This conversion ensures that all units are consistent, preventing any calculation mishaps.

• Speed in km/h
• Time in hours
• Distance in kilometers

This attention to detail is crucial for accurately finding how far the kangaroo hops.

Distance Calculation

Calculating distance in kinematics involves using the basic formula \(\text{Distance} = \text{Speed} \times \text{Time}\). Once you have converted all units to be the same format, you can plug these values into the formula. For the red kangaroo, hopping at 65 km/h for 0.0533 hours, you multiply the speed by the time to get the distance: \[ 65 \text{ km/h} \times 0.0533 \text{ hours} = 3.4645 \text{ kilometers} \]

This simple formula is versatile. Whether you're determining how far something travels or how long it takes to travel a given distance, understanding each component of this formula can help simplify problem-solving in kinematics. In contrast, if you're given the distance and need to find time or speed, rearranging this formula is easy once you understand the basics.