Physics - Understanding Distance, Velocity, and Acceleration

While traveling along a highway, a specific automobile is capable of an acceleration of about $1.6m/s^{2}$ . At this rate, how long does it take to accelerate from to ?

Explanation

Knowns:

$a = 1.6m/s^2{}$

$v_{0}= 65km/h$

$v_{f}= 120km/h$

Unknowns:

$\Delta t = ?$

Equation:

$a=\frac{{}v_{f} - v_{0}}{\Delta t}$

The most important thing to recognize here is that the initial and final velocities are not in the correct units. The first step is to convert both of these values to .

$\frac{65km}{h} * \frac{1h}{60 min} * \frac{1 min}{60 s} *\frac{100m}{km} = 18.06m/s$

$\frac{120km}{h} * \frac{1h}{60 min} * \frac{1 min}{60 s} *\frac{100m}{km} = 33.33m/s$

Then rearrange your equation to solve for (the missing variable).

${\Delta t}=\frac{{}v_{f} - v_{0}}{a}$

Now plug in the variables and solve.

${\Delta t}=\frac {33.33m/s - 18.06m/s}{1.6m/s^2}$

$\Delta t = 9.55s$

An athlete kicks a ball into the air. It travels $\small 42m$ and is in the air for $\small 4.5s$ . How fast must the athlete run from the point where he kicks the ball in order to catch it before it lands?

$9.33\frac{m}{s}$

$10.21\frac{m}{s}$

$9.19\frac{m}{s}$

$18.7\frac{m}{s}$

$8.74\frac{m}{s}$

Explanation

To solve this problem, we need to understand what speed is. Speed is the distance covered in a given amount of time.

$\text{speed}=\frac{\text{distance}}{\text{time}}$

In our problem, we need to find the speed of the athlete. We are given the distance the athlete must cover, which is equal to the distance traveled by the ball.

We are also told how much time the athlete has to cover the distance, which is equal to the time the ball is in the air.

Use these values and the equation for speed to find the speed that the athlete must run.

$s=\frac{d}{t}=\frac{42m}{4.5s}$

$s=9.33\frac{m}{s}$

During a storm, you can usually see the lightning before you hear the thunder, unless you are very close to the lightning strike. What causes this discrepancy?

The speed of light is much faster than the speed of sound

The speed of sound is much faster than the speed of light

There is no definitive scientific reason for this phenomenon

We need to know the voltage of the lightning in order to determine the answer

We need to know the current of the lightning in order to determine the answer

Explanation

Assuming you stand in one place, the distance between you and the lightning strike does not change.

The formula for velocity is:

$v=\frac{\Delta x}{\Delta t}$

In this scenario, the distance travelled, $\Delta x$ , does not change. The time taken to travel this distance, $\Delta t$ , does change. That means that the velocity must also be changing.

This is an indirect relationship. As $\Delta t$ increases, will decrease; thus, the object with a greater time of travel (sound) will have a slower velocity.

Two locomotives approach each other on parallel tracks. They both have the same speed of with respect to ground. If they are initially apart, how long will it be before they reach each other?

Explanation

Knowns:

Unknowns:

Equation:

Since both trains are traveling at a constant velocity and there is no acceleration the equation is

$v = \frac{\Delta x}{\Delta t}$

Rearrange this equation to solve for time.

$\Delta t = \frac{\Delta x}{v}$

Since both trains are traveling at the same speed, the trains will meet up in the middle of the total distance between the trains.

$\frac{Total Distance}{2}= \frac{7.2km}{2} = 3.6km$

Plug this distance into the equation to find the time that it takes to travel the distance.

$\Delta t = \frac{3.6km}{120km/h}$

$\Delta t = 0.3 h$

The answers provided are in minutes, so the final step is to convert the time to minutes.

$0.3h * \frac{60min}{1h}= 1.8 min$

An Olympic sprinter runs a dash in . Assuming he starts at rest, what is his average velocity?

$10.3\frac{m}{s}$

We need to know the acceleration in order to solve.

$972\frac{m}{s}$

$1.06\frac{m}{s}$

$9.8\frac{m}{s}$

Explanation

The average velocity is the total displacement divided by the total time or $v=\frac{\Delta x}{\Delta t}$ .

Plug in our given values.

$v=\frac{\Delta x}{\Delta t}$

$v=\frac{100m}{9.72s}$

$v=10.3\frac{m}{s}$

Which of the following is a correct measurement of velocity in SI units?

$3\frac{m}{s}\ \text{to the right}$

$4mph\ \text{East}$

$3\frac{m}{s}$

$3\frac{m}{hr}\ \text{upward}$

Explanation

Velocity is a vector measurement of displacement per unit time. This means that it has both magnitude and direction, according to the direction of the displacement.

$\vec{v}=\frac{\vec{d}}{t}$

The standard unit for displacement is meters and the standard unit for time is seconds; thus, the SI unit for velocity is meters per second. Any standard measure of velocity will be given in meters per second, with a corresponding direction of action.

A bowling ball traveling with constant speed hits the pins at the end of a bowling lane long. The bowler hears the sound of the ball hitting the pins after the ball is released from his hands. What is the speed of the ball, assuming the speed of sound is ?

Explanation

Knowns:

$\Delta x = 18.3m$

$\Delta t_{total} = 3.1s$

$v_{sound}=340m/s$

Unknowns:

$v_{ball}= ?$

Time can be broken up as well. There is a time that it takes the ball to roll down the lane to hit the pin. There is also a different time for the sound to reach the person’s ear. This adds up to the total time provided in the problem.

$\Delta t_{ball}= ?$

$\Delta t_{sound}= ?$

Equation:

The ball and sound both travel at constant velocities. Therefore the equation that can best be used is

$v=\frac{\Delta x}{\Delta t}$

Each step must be taken into account as the ball travels down the alley and as the sound travels back. Since the velocity of the ball is not provided, the best place to start is to find the amount of time that it takes for the sound to reach the person’s ear.

Rearrange the equation to solve for time.

$\Delta t = \frac{\Delta x}{v}$

Plug in the values for the speed of sound and the distance that the sound travels.

$\Delta t = \frac{18.3m}{340m/s}$

$\Delta t = 0.06s$

The total time was given. Subtract the time that it takes the sound to travel to determine how long the ball was rolling down the lane.

$\Delta t = 3.1s -0.054s$

$\Delta t =3.046s$

Now, use this time in the equation with the distance of the bowling to determine the velocity of the ball

$v=\frac{18.3m}{3.046s}$

One of the fastest players in football ran in . What was his average speed during this time?

$8.63\frac{m}{s}$

$9.43\frac{m}{s}$

$1.16\frac{m}{s}$

$155\frac{m}{s}$

$19.2\frac{m}{s}$

Explanation

To solve this problem we need to consider the definition of speed, which is the distance traveled over a given amount of time.

Even though the player's speed is changing throughout the sprint (due to acceleration), we are asked to find the average speed. We can do this using the total distance and total time given. The distance is and the time is .

$speed=\frac{distance}{time}$

$speed=\frac{36.6m}{4.24s}$

$speed=8.63\frac{m}{s}$

You are given a graph of displacement vs. time. Which of the following ways can be used to determine velocity at any given point?

The velocity is the slope of the line at any given time

The velocity is the area under the curve for any time interval

The velocity is the total change in displacement over the total time

The velocity is the y-intercept of the graph

The velocity cannot be determined from a graph of displacement vs time

Explanation

The important thing to note is that the question asks for the velocity at any given point. The average velocity will be equal to the total displacement divided by the total time, but the question is asking for the instantaneous velocity.

Velocity is calculated by a change in displacement over a change in time. In a displacement versus time graph, this is equal to the slope.

$\text{slope}=\frac{y_2-y_1}{x_2-x_1}=\frac{d_2-d_1}{t_2-t_1}=\frac{\Delta d}{\Delta t}$

$v=\frac{\Delta d}{\Delta t}$

This tells us that the slope at a certain time will be equal to the velocity at that time.

In calculus terms, velocity is the derivative of the function for displacement in terms of time. What this means is that at any point on the graph, the instantaneous slope is the velocity for that given time. To determine velocity, one must find the slope of the line at that particular time interval.

One of the fastest players in football ran in . What was his average speed during this time?