Question

A \(523-\mathrm{kg}\) experimental rocket sled can be accelerated from rest to \(1620 \mathrm{~km} / \mathrm{h}\) in \(1.82 \mathrm{~s}\). What net force is required?

Short answer

The net force required to accelerate the rocket sled is approximately \(129308 \, \text{N}\).

Step-by-step solution

Identifying and Converting Units

Identify the given variables: mass (m) = 523 kg, initial velocity (u) = 0 km/h, final velocity (v) = 1620 km/h, and time (t) = 1.82 s. For consistent units in the upcoming calculations, convert the final velocity from km/h to m/s: \(1620 \, \text{km/h} = 1620 \times \, \frac{1000}{3600} \, \text{m/s} = 450 \, \text{m/s}\).

Calculating Acceleration

Accelerations (a) is the change in velocity per unit time. The formula is \(a = \frac{v - u}{t}\). Substituting the values, we get: \(a = \frac{450 \, \text{m/s} - 0}{1.82 \, \text{s}} = 247.25 \, \text{m/s}^2\).

Calculating Force

Force (F) can be calculated using Newton's second law of motion: \(F = m \times a\). This gives: \(F = 523 \, \text{kg} \times 247.25 \, \text{m/s}^2 = 129307.75 \, \text{N}\).

Key concepts

Force Calculation

Newton's Second Law of Motion is key to understanding force calculation. It is expressed as:
\[ F = m \times a \]where:

• \( F \) is the net force applied, measured in Newtons (N)
• \( m \) is the mass of the object in kilograms (kg)
• \( a \) is the acceleration in meters per second squared (m/s²)

To determine the force required to accelerate an object, multiply its mass by the acceleration it experiences. Remember, the force is the cause of the acceleration sensed by the object. So, in our example, a 523 kg rocket sled accelerating at 247.25 m/s² requires a force of 129,307.75 N. This means a mighty push is necessary to overcome inertia and speed up the sled that quickly.

Unit Conversion

In physics, consistency in units simplifies calculations and prevents errors. Often, problems provide measurements in various units, requiring conversion to be compatible with equations. One common conversion is between kilometers per hour (km/h) and meters per second (m/s). Using the relation:
\[ 1 \, \text{km/h} = \frac{1000}{3600} \, \text{m/s} = \frac{5}{18} \, \text{m/s} \]Apply this to convert final velocities from km/h to m/s. So, a velocity of 1620 km/h becomes 450 m/s after conversion. These conversions ensure that all variables are expressed in compatible units, crucial for correct application of physics formulas, like those in Newton's laws.

Acceleration

Acceleration is the rate at which an object's velocity changes over time. It is defined by the equation:
\[ a = \frac{v - u}{t} \]where:

• \( v \) is the final velocity
• \( u \) is the initial velocity
• \( t \) is the time period over which the change occurs

In our scenario, the initial velocity is 0 m/s, and the final velocity is 450 m/s, achieved over 1.82 seconds. Substituting these values, the acceleration is calculated as 247.25 m/s². High acceleration indicates how rapidly the rocket sled increases speed. Understanding acceleration helps grasp how quickly a body can change its state of motion.