Question

(a) What is the magnitude of the momentum of a 10,000-kg truck whose speed is 12.0 m/s? (b) What speed would a 2000-kg SUV have to attain in order to have (i) the same momentum? (ii) the same kinetic energy?

Short answer

(a) 120,000 kg·m/s; (b-i) 60 m/s; (b-ii) 26.83 m/s.

Step-by-step solution

Understanding Momentum

Momentum is defined as the product of an object's mass and its velocity. The formula is given by: \( p = m \times v \), where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity.

Calculating Truck's Momentum

To find the momentum of the truck, substitute the given values into the momentum equation: the mass \( m = 10,000 \) kg and the speed \( v = 12.0 \) m/s. Thus, the momentum \( p = 10,000 \times 12.0 = 120,000 \) kg·m/s.

Setting SUV Momentum Equal to Truck's

For the SUV to have the same momentum as the truck, we set its momentum equation equal to the truck's: \( m_{SUV} \times v_{SUV} = 120,000 \). The mass of the SUV \( m_{SUV} = 2,000 \) kg, so \( 2,000 \times v_{SUV} = 120,000 \).

Solving for SUV's Speed with Equal Momentum

Divide both sides of the equation \( 2,000 \times v_{SUV} = 120,000 \) by 2,000 to find \( v_{SUV} \). Thus, \( v_{SUV} = \frac{120,000}{2,000} = 60 \) m/s.

Understanding Kinetic Energy

Kinetic energy is expressed by the formula \( KE = \frac{1}{2} m v^2 \), where \( KE \) is the kinetic energy, \( m \) is mass, and \( v \) is velocity.

Calculating Truck's Kinetic Energy

Calculate the kinetic energy of the truck using its mass and velocity: \( KE = \frac{1}{2} \times 10,000 \times (12.0)^2 = 720,000 \) J.

Setting SUV's Kinetic Energy Equal to Truck's

For the SUV to have the same kinetic energy as the truck, the equation becomes: \( \frac{1}{2} \times 2,000 \times v_{SUV}^2 = 720,000 \).

Solving for SUV's Speed with Equal Kinetic Energy

First simplify \( \frac{1}{2} \times 2,000 \times v_{SUV}^2 = 720,000 \) to \( 1,000 \times v_{SUV}^2 = 720,000 \). Then divide by 1,000: \( v_{SUV}^2 = 720 \). Taking the square root gives \( v_{SUV} = \sqrt{720} \approx 26.83 \) m/s.

Key concepts

Kinetic Energy

Kinetic energy, frequently abbreviated as KE, refers to the energy an object possesses due to its motion. To calculate kinetic energy, you use the formula: \( KE = \frac{1}{2} m v^2 \). This involves two significant factors: the mass \( m \) of the object and its velocity \( v \).
A larger mass or a higher speed results in greater kinetic energy. This is why heavier vehicles or faster-moving objects have greater energy when they're in motion.
The interplay of mass and velocity in the kinetic energy equation highlights how both aspects contribute significantly to the energy an object holds while moving. This understanding is central when comparing different vehicles like our truck and SUV scenario, where we aim to equalize kinetic energy between two differently sized vehicles.

Mass and Velocity Relationship

The relationship between mass and velocity is crucial to understanding how both contribute to momentum and kinetic energy. In physics, mass is considered a scalar quantity, meaning it has only magnitude. On the other hand, velocity is a vector quantity with both magnitude and direction. Therefore, when calculating momentum and kinetic energy, the product of these gives us net results that define the movement and energy characteristics of an object.
With momentum expressed as \( p = m \times v \), a heavier object with the same speed as a lighter one packs more momentum. But what if we wish to keep momentum constant? Adjusting mass means velocity must inversely change. A smaller mass requires a higher velocity to maintain the same momentum, as shown in the SUV and truck problem.

Momentum Calculation

Momentum is a fundamental concept in physics, often described as the "quantity of motion" of an object. It's determined by multiplying the object's mass by its velocity, given by the equation \( p = m \times v \). The unit used for momentum is kg·m/s.
In our truck and SUV exercise, the momentum of the 10,000-kg truck at 12 m/s was 120,000 kg·m/s. To find what speed a 2,000-kg SUV would need to match this momentum, we set its momentum calculation equal to the truck's, as the problem involves working with the principle that momentum \( p \) should be equal to ensure similarity in dynamic impact.

Physics Problem-Solving

Solving physics problems involves applying fundamental principles and formulas systematically. In scenarios involving motion, understanding the relationships and interdependencies between concepts like momentum and kinetic energy is crucial.
Start by identifying given values and required unknowns. Use relevant formulas, ensuring units are consistent. Rearrange the equations where necessary, substituting the values methodically to solve for unknowns.
In the momentum and kinetic energy problem, this involved setting the truck's values and deducing the SUV's necessary speeds using momentum and kinetic energy equivalence. Breaking down problems into steps makes complex scenarios more manageable and highlights the relationship between different physical properties.

SUV Speed Calculation

Calculating the speed of the SUV involved setting and solving two separate scenarios based on momentum and kinetic energy equivalence.
For momentum, we used the equation \( 2,000 \times v_{SUV} = 120,000 \) to solve for \( v_{SUV} \). Dividing both sides by 2,000 gave us a speed of 60 m/s.
For kinetic energy, we required the equation \( \frac{1}{2} \times 2,000 \times v_{SUV}^2 = 720,000 \). Simplifying this equation revealed \( v_{SUV}^2 = 720 \). The square root of 720 provided a speed approximately equal to 26.83 m/s.
The calculations highlight how different physics principles can predict varying speeds for equivalent dynamic properties in different mass vehicles.