Question

(a) What is the velocity of an object that moves from \(73 \mathrm{~m}\) to \(62 \mathrm{~m}\) in \(12 \mathrm{~s}\) ? (b) What is its speed?

Short answer

The velocity is \(-0.9167 \,\text{m/s}\) and the speed is \(0.9167 \,\text{m/s}\).

Step-by-step solution

Identify Initial and Final Positions

First, identify the initial position \( x_i \) and the final position \( x_f \) of the object. Here, \( x_i = 73 \,\text{m} \) and \( x_f = 62 \,\text{m} \).

Calculate Displacement

Calculate the displacement \( \Delta x \) of the object using the formula \( \Delta x = x_f - x_i \). Substitute the values to get \( \Delta x = 62 \,\text{m} - 73 \,\text{m} = -11 \,\text{m} \).

Determine Time Interval

Identify the time interval \( \Delta t \) during which the displacement occurred. In this case, \( \Delta t = 12 \,\text{s} \).

Calculate Velocity

The velocity \( v \) is given by the displacement divided by the time interval: \( v = \frac{\Delta x}{\Delta t} \). Substitute \( \Delta x = -11 \,\text{m} \) and \( \Delta t = 12 \,\text{s} \) to calculate \( v = \frac{-11 \,\text{m}}{12 \,\text{s}} = -0.9167 \,\text{m/s} \).

Calculate Speed

Speed is the magnitude of velocity. Therefore, speed is the absolute value of velocity. For this problem, the speed is \( |v| = |-0.9167 \,\text{m/s}| = 0.9167 \,\text{m/s} \).

Key concepts

Displacement

When we talk about displacement, we're referring to the change in position of an object. It's how far out of place an object is; it's the object's overall change in position. Displacement can be a bit tricky because it's a vector quantity. This means it has both magnitude and direction. Knowing this helps us understand why displacement can be negative.

In our exercise, the object moves from an initial position of 73 meters to a final position of 62 meters. To find the displacement, we use the formula:

• \( \Delta x = x_f - x_i \)

Plugging in the values, we have:

• \( \Delta x = 62 \, \text{m} - 73 \, \text{m} = -11 \, \text{m} \)

The displacement is -11 meters because the object moved towards a lower position, or simply put, it moved backwards by 11 meters.

Time Interval

The time interval is a key concept when we want to measure how long an event takes place. It's a measure of the duration from the start to the finish of the event. This is straightforward, as it's a scalar quantity, meaning it only has magnitude and no direction.

For our exercise, the displacement occurs over a period of 12 seconds. Thinking of time intervals as the lengths of time parts of an action take can make it easier to grasp. Here, we record:

• \( \Delta t = 12 \, \text{s} \)

By knowing this, we are equipped to calculate other important values like velocity by pairing it with displacement.

Speed Calculation

Speed is all about how fast something is moving, without considering the direction. Like time intervals, speed is a scalar quantity. When calculating speed, we look at the absolute value of velocity, which gives us only how fast the object is moving regardless of direction.

For calculating the velocity, which is the rate of change of displacement, we use the equation:

• \( v = \frac{\Delta x}{\Delta t} \)

Substituting in our values:

• \( v = \frac{-11 \, \text{m}}{12 \, \text{s}} = -0.9167 \, \text{m/s} \)

The negative sign indicates the direction of the velocity, backwards in this case. But for speed, we take the absolute value:

• Speed = \(|-0.9167 \, \text{m/s}| = 0.9167 \, \text{m/s}\)

So, speed tells us how fast the object was moving, irrespective of which direction it was traveling.