Q: Step 1: Calculate the initial kinetic energy.

Using our standard formula: $KE_i = 0.5 \times 1,000 \text{ kg} \times (20 \text{ m/s})^2$. $KE_i = 0.5 \times 1,000 \times 400 = 200,000$ Joules.

Q: Step 4: Use the Work formula to find the distance.

We know $W = F \times d$. We know the Work required is 200,000 J, and the Force applied by the brakes is 8,000 N. $200,000 = 8,000 \times d$. To solve for distance ($d$), divide the Work by the Force: $200,000 / 8,000 = 25$. The car will skid for exactly 25 meters before coming to a complete halt. This calculation is the exact mathematics used by accident reconstructionists to determine how fast a car was speeding based on the length of the skid marks left on the asphalt.

Question 1

Step 1: Identify and format the variables.

Accepted Answer

Ensure all your numbers are in standard SI units. 
Mass ($m$) = 1,500 kg. 
Velocity ($v$) = 30 m/s. 
Both are in the correct units, so no conversion is necessary.

Question 2

Step 2: Square the velocity.

Accepted Answer

According to the order of operations, exponents come first. You must multiply the velocity by itself.
$v^2 = 30 	imes 30 = 900$.
The unit also squares, becoming meters squared per second squared ($	ext{m}^2/	ext{s}^2$).

Question 3

Step 3: Multiply by the mass.

Accepted Answer

Take the squared velocity and multiply it by the mass of the car.
$1,500 	ext{ kg} 	imes 900 	ext{ m}^2/	ext{s}^2 = 1,350,000$.

Question 4

Step 4: Multiply by one-half.

Accepted Answer

Finally, multiply the result by 0.5 (or divide by 2) to complete the formula.
$1,350,000 	imes 0.5 = 675,000$.

Question 5

Step 1: Calculate the initial kinetic energy.

Accepted Answer

Using our standard formula: $KE_i = 0.5 	imes 1,000 	ext{ kg} 	imes (20 	ext{ m/s})^2$.
$KE_i = 0.5 	imes 1,000 	imes 400 = 200,000$ Joules.

Question 6

Step 2: Determine the final kinetic energy.

Accepted Answer

Because the car is coming to a complete stop, its final velocity is 0 m/s. Therefore, its final kinetic energy ($KE_f$) is 0 Joules.

Question 7

Step 3: Calculate the change in kinetic energy.

Accepted Answer

$\Delta KE = KE_f - KE_i = 0 - 200,000 = -200,000$ Joules. The negative sign simply indicates that energy is being removed from the car. The total work required to stop the car is 200,000 Joules.

Question 8

Step 4: Use the Work formula to find the distance.

Accepted Answer

We know $W = F 	imes d$. We know the Work required is 200,000 J, and the Force applied by the brakes is 8,000 N.
$200,000 = 8,000 	imes d$.
To solve for distance ($d$), divide the Work by the Force: $200,000 / 8,000 = 25$.
The car will skid for exactly 25 meters before coming to a complete halt. This calculation is the exact mathematics used by accident reconstructionists to determine how fast a car was speeding based on the length of the skid marks left on the asphalt.

Question 9

Can kinetic energy ever be a negative number?

Accepted Answer

No, kinetic energy can never be negative in classical mechanics. The formula is $\frac{1}{2} m v^2$. Mass ($m$) is an absolute measure of matter and cannot be negative. Velocity ($v$) can be negative if an object is moving in a negative direction on a coordinate plane, but because the velocity is squared in the formula, any negative number multiplied by itself becomes a positive number. Therefore, the lowest possible kinetic energy an object can possess is exactly zero, which occurs when it is completely at rest.

Question 10

How does doubling the speed of an object affect its kinetic energy?

Accepted Answer

Doubling the speed of an object will quadruple its kinetic energy. This is because velocity is squared in the kinetic energy formula ($v^2$). If a car is moving at 10 m/s, its velocity factor is $10^2 = 100$. If the car accelerates to 20 m/s (double the speed), its velocity factor becomes $20^2 = 400$. The energy has increased by a factor of four. This exponential relationship is the primary reason why high-speed car crashes are exponentially more lethal than low-speed crashes.

Question 11

Is kinetic energy always conserved during a collision between two objects?

Accepted Answer

No, kinetic energy is rarely conserved in real-world collisions. In physics, a collision where kinetic energy is perfectly conserved is called an "elastic collision" (like two perfectly rigid billiard balls bouncing off one another). However, most real-world collisions are "inelastic." In a car crash, the total momentum is conserved, but a massive amount of kinetic energy is permanently lost from the mechanical system. It is converted into the work required to permanently deform the metal chassis, the thermal energy of heat generated by friction, and the acoustic energy of the sound of the crash.

Question 12

What is the difference between kinetic energy and momentum?

Accepted Answer

While both involve mass and velocity, they are fundamentally different. Momentum ($p = mv$) is a vector quantity (it has direction) that measures the "quantity of motion" and is conserved in all collisions. It dictates how hard it is to stop an object. Kinetic energy ($KE = \frac{1}{2} m v^2$) is a scalar quantity (no direction) that measures the "energy of motion" and the capacity to do work. Because velocity is squared in kinetic energy, a light, fast object (like a bullet) can have very little momentum but a massive amount of kinetic energy.

Question 13

How do you calculate kinetic energy if you only know the momentum and the mass?

Accepted Answer

You can calculate kinetic energy directly from momentum using an alternative formula: $KE = \frac{p^2}{2m}$, where $p$ is momentum and $m$ is mass. This is derived algebraically by taking the standard formula ($KE = \frac{1}{2} m v^2$) and multiplying the top and bottom by mass ($m$). This gives you $\frac{m^2 v^2}{2m}$. Since momentum $p = mv$, the numerator $m^2 v^2$ is exactly equal to $p^2$. This shortcut is incredibly useful in advanced physics problems where velocity is not explicitly provided.

Question 14

Does kinetic energy depend on the direction the object is moving?

Accepted Answer

No, kinetic energy is strictly a scalar quantity, meaning it is entirely independent of direction. It only depends on the magnitude of the velocity (the speed). A 1,000 kg car traveling North at 20 m/s has exactly the same kinetic energy (200,000 Joules) as an identical car traveling South, East, or West at 20 m/s. If the two cars collide head-on, their momentums (being vectors) will cancel each other out, but their kinetic energies will add together, resulting in a 400,000 Joule impact.

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