Question

identify the values of the variables needed to find the kinetic energy in each example. then, calculate the kinetic energy of the moving objects.
a roller coaster with 6 people on board has a total mass of 200 kg. it slowly rises to 50 m above the ground. as it drops, it reaches a top speed of 20 m/s.

m =
v =
e_k =

an 8 m cannon can fire a 15 kg cannonball every 60 seconds. using 100 g of gunpowder for each blast, the cannonball can reach a maximum speed of 25 m/s.

m =
v =
e_k =

Explanation:

Step1: Identify variables for Roller Coaster

The formula for kinetic energy is \( E_k = \frac{1}{2}mv^2 \). For the roller coaster:

• Mass (\( m \)): 200 kg (given as total mass)
• Velocity (\( v \)): 20 m/s (given as top speed)
• Kinetic Energy (\( E_k \)): To be calculated.

Step2: Calculate Kinetic Energy for Roller Coaster

Substitute \( m = 200 \, \text{kg} \) and \( v = 20 \, \text{m/s} \) into the formula:

$$ E_k = \frac{1}{2} \times 200 \times (20)^2 $$

First, calculate \( (20)^2 = 400 \). Then:

$$ E_k = \frac{1}{2} \times 200 \times 400 = 100 \times 400 = 40000 \, \text{J} $$

Step3: Identify variables for Cannonball

For the cannonball:

• Mass (\( m \)): 15 kg (given mass of cannonball)
• Velocity (\( v \)): 25 m/s (given maximum speed)
• Kinetic Energy (\( E_k \)): To be calculated.

Step4: Calculate Kinetic Energy for Cannonball

Substitute \( m = 15 \, \text{kg} \) and \( v = 25 \, \text{m/s} \) into the formula:

$$ E_k = \frac{1}{2} \times 15 \times (25)^2 $$

First, calculate \( (25)^2 = 625 \). Then:

$$ E_k = \frac{1}{2} \times 15 \times 625 = 7.5 \times 625 = 4687.5 \, \text{J} $$

Answer:

• Roller Coaster: \( m = 200 \, \text{kg} \), \( v = 20 \, \text{m/s} \), \( E_k = 40000 \, \text{J} \)
• Cannonball: \( m = 15 \, \text{kg} \), \( v = 25 \, \text{m/s} \), \( E_k = 4687.5 \, \text{J} \)