Problem 14 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A sled is initially given a shove up a frictionless \(23.0^{\circ}\) incline. It reaches a maximum vertical height \(1.12 \mathrm{~m}\) higher than where it started. What was its initial speed?

Step-by-Step Solution

Answer

The initial speed of the sled was approximately 4.684 m/s.

Step 1: Understanding the problem

We need to find the initial speed of a sled given that it climbs a frictionless incline with an initial shove and reaches a maximum vertical height of 1.12 meters.

Step 2: Utilizing Energy Conservation

According to the law of conservation of energy, the initial kinetic energy of the sled should equal the potential energy at the maximum height. The equation to express this is \( \frac{1}{2} mv^2 = mgh \), where \( m \) is mass, \( v \) is the initial velocity, \( g \) is the acceleration due to gravity (9.8 m/s²), and \( h \) is the height (1.12 m).

Step 3: Cancel out mass and solve for velocity

Since mass \( m \) appears on both sides of the equation, it can be canceled out. We then solve for \( v \). The equation simplifies to \( \frac{1}{2} v^2 = gh \). Rearranging for \( v \), we get \( v = \sqrt{2gh} \).

Step 4: Substitute known values

Substitute \( g = 9.8 \, \text{m/s}^2 \) and \( h = 1.12 \, \text{m} \) into the expression \( v = \sqrt{2gh} \) which results in \( v = \sqrt{2 \times 9.8 \times 1.12} \).

Step 5: Calculation

Calculate \( v = \sqrt{2 \times 9.8 \times 1.12} \). This results in \( v = \sqrt{21.952} \approx 4.684 \text{ m/s} \).

Key Concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. It can be thought of as the energy that keeps things moving.
This type of energy depends on the mass of the object and its velocity. The formula to calculate kinetic energy is given by \( KE = \frac{1}{2} m v^2 \), where:

• \( m \) is the mass of the object
• \( v \) is the velocity of the object

The more mass an object has or the faster it moves, the more kinetic energy it has.
Consider a sled going uphill; the initial push gives it a velocity, endowing it with kinetic energy. This energy propels the sled up the incline.This is crucial because, on a frictionless incline, the sled needs enough kinetic energy to reach a given height or it might not make it.

Potential Energy

Potential energy is the stored energy in an object due to its position or state. In the context of an inclined plane, it’s the energy an object has when it is elevated.
The important equation for gravitational potential energy is \( PE = mgh \), where:

• \( m \) is the mass of the object
• \( g \) is the acceleration due to gravity (approximately \(9.8 \, \text{m/s}^2\) on Earth)
• \( h \) is the height above the ground

Potential energy increases as an object ascends the incline. This is due to the height \( h \) increasing as the sled is pushed upwards.
Thus, reaching a maximum vertical height means the sled's potential energy is at its peak, ready to be converted back into kinetic energy when descending or to do other work.

Inclined Plane

An inclined plane is a flat surface tilted at an angle. It’s a simple machine that aids in moving objects to a different elevation.
Understanding inclined planes involves some geometry and physics principles. The plane allows a smoother conversion of energies.In scenarios involving conservation of energy, motion on inclined planes is especially insightful.
On a frictionless incline, like in our sled problem, energy transitions smoothly between kinetic and potential forms. The slope affects this transition. The angle of inclination, in this case, \(23^{\circ}\), plays a role in determining the height gained or the distance traveled. As the sled moves up the incline, its kinetic energy decreases while potential energy increases, until it stops momentarily before sliding back down.