Question

using kinetic energy to determine height
read the scenario and solve these two problems.
when traveling at top speed, a roller coaster train with a mass of 12,000 kg has a velocity of 30 m/s. the kinetic energy of the train at top speed is ◯ j.
given this kinetic energy, what is the tallest hill this roller coaster train can reach the top of?
the train can climb a hill that is ◯ m high.

Explanation:

First Problem: Calculate Kinetic Energy

Step1: Recall Kinetic Energy Formula

The formula for kinetic energy (KE) is $KE = \frac{1}{2}mv^2$, where $m$ is mass and $v$ is velocity.

Step2: Substitute Values

Given $m = 12000\ kg$ and $v = 30\ m/s$. Substitute into the formula:
$KE = \frac{1}{2} \times 12000 \times (30)^2$

Step3: Calculate

First, calculate $(30)^2 = 900$. Then, $\frac{1}{2} \times 12000 = 6000$. Multiply: $6000 \times 900 = 5400000\ J$.

Second Problem: Determine Maximum Height

Step1: Use Conservation of Energy

At the top of the hill, kinetic energy is converted to gravitational potential energy ($PE = mgh$), so $KE = PE$. Thus, $mgh = KE$.

Step2: Solve for Height ($h$)

Rearrange the formula: $h = \frac{KE}{mg}$. We know $KE = 5400000\ J$, $m = 12000\ kg$, and $g = 9.8\ m/s^2$ (acceleration due to gravity).

Step3: Substitute and Calculate

$h = \frac{5400000}{12000 \times 9.8}$
First, calculate $12000 \times 9.8 = 117600$. Then, $h = \frac{5400000}{117600} \approx 45.92\ m$ (or using $g = 10\ m/s^2$ for approximation: $h = \frac{5400000}{12000 \times 10} = 45\ m$).

Answer:

First problem: $\boldsymbol{5400000}$ J
Second problem: Approximately $\boldsymbol{45.92}$ m (or $\boldsymbol{45}$ m if using $g = 10\ m/s^2$)