The relationship between frequency and period is:

We are given the period in the question. Using this value, we can solve for the frequency.

The question asks to point out the false statement. Everything on Earth is accelerated downwards by gravity, all the time, by . Think of gravity as having a negative sign. When the ball is thrown up, acceleration is working against the velocity slowing the ball down. Their signs are opposite. But when the ball is falling back down to Earth, the velocity and acceleration have the same sign. So velocity and acceleration will not always have the same sign.

If gravity could be said to have a negative sign since it pulls everything downward, then an upward velocity would have a positive (and opposing sign). At the top of the trajectory when the ball's upward velocity is finally overcome by gravity, the sign of the velocity becomes negative as it now points back down to Earth. So it is true velocity changes sign at the top.

Since the ball is caught at the same height, both the time up and time down are equal and it will be traveling at the same speed. Since gravity is the only force acting on it, the ball loses all its velocity on the way up and regains that exact amount by the time it reaches the height it started the journey. This also makes the time up equal to the time it falls. If the same force acts with same strength (gravity) the entire time, why would either of these change?

The velocity of a wave is equal to the product of the wavelength and frequency:

We can rearrange this formula to solve for the wavelength.

We also know that the period is the inverse of the frequency:

Substitute this into the equation for wavelength.

Now we can use our given values for the period and the velocity to solve for the wavelength.

The speed of the wave along the Slinky depends on the mass of the Slinky itself and the tension caused by stretching it. Since both of these things have not changed, the wave speed remains constant.

The wave speed is equal to the wavelength multiplied by the frequency.

Since she is moving her hand faster, the frequency has increased. Since the velocity has not changed, an increase in the frequency would decrease the wavelength.

We also can calculate the total number of seconds in a day.

There are seconds in one day.

Therefore we want our clock to swing a certain number of times with a period of to equal .

We know that our current clock has a certain number of swings with a period of to equal

So we have

We can calculate the current period of the pendulum using the equation

We can set up a ratio of each of these two periods to determine the missing length.

Notice that 2, pi and g are all in both the numerator and denominator and therefore fall out of the problem.

We can now solve for our missing piece.

Square both sides to get rid of the square root.

We should lengthen the pendulum by

To solve this problem, recognize that the force due to friction must equal the centripetal force of the curve:

This will give the maximum force that the car can have in the curve without skidding. Expand the equation for centripetal force.

We are given the value for the force of friction, the mass of the car, and the radius of the curve. Using these values, we can find the velocity.

The relationship between velocity, frequency, and wavelength is:

In this case we're given a scenario where and . The velocities equal each other because the problem states it has a constant velocity. Therefore we can set these equations equal to each other:

Notice that the 's cancel out:

Divide both sides by two:

Frequency is equal to the reciprocal of the period:

Given the period, we can invert it to find the frequency.

For this problem, we'll use Newton's third law, which states that for every force there will be another force equal in magnitude, but opposite in direction.

This means that the force of the first skater on the second will be equal in magnitude, but opposite in direction:

Use Newton's second law to expand this equation.

We are given the mass of each skater and the acceleration of the first. Using these values, we can solve for the acceleration of the second.

From here, we need to isolate the acceleration of the second skater.

Notice that the acceleration of the second skater is negative. Since she is moving in the opposite direction of the first skater, one acceleration will be positive while the other will be negative as acceleration is a vector.

The relationship between frequency and wavelength determines the velocity:

The frequency is the inverse of the period. We can substitute this into the equation above.

In the question, both of the notes are played at the same time in the same location, so they both should have the same velocity. We can set the equation for each tone equal to each other.

We are told that . Substitute into our equation.

We can cancel the period from each side of the equation, leaving the relationship between the two wavelengths.

The wavelength of the first wave is equal to half the wavelength of the second. This means that the wavelength for the tone with a longer period will have a longer wavelength as well.