Question

A projectile with a mass of \(2.40 \mathrm{~kg}\) is fired from a cliff \(125 \mathrm{~m}\) high with an initial velocity of \(150 \mathrm{~m} / \mathrm{s}\), directed \(41.0^{\circ}\) above the horizontal. What are \((a)\) the kinetic energy of the projectile just after firing and \((b)\) its potential energy? \((c)\) Find the speed of the projectile just before it strikes the ground. Which answers depend on the mass of the projectile? Ignore air drag.

Short answer

The kinetic energy just after firing is 27000 J. The potential energy is 29412 J. The speed of the projectile just before it strikes the ground is 151.4m/s. Both kinetic and potential energies depend on the mass of the projectile while the final speed does not.

Step-by-step solution

Calculate the kinetic energy just after firing

To find out the kinetic energy right after the firing, the following formula can be used:\n Kinetic energy = 0.5 * mass * (velocity)^2. Here, mass equals 2.4kg and velocity equals 150 m/s. Substituting these values into the formula yields:\n Kinetic energy = 0.5 * 2.4kg * (150m/s)^2.

Calculate the potential energy

To calculate potential energy, the formula potential energy = mass * g * height is used, where g is gravity (approx. 9.81 m/s^2) and height is the distance above the ground (125m). So, the potential energy = 2.4kg * 9.81m/s^2 * 125m.

Calculate the speed of the projectile just before it hits the ground

The initial vertical and horizontal velocities can be calculated using trigonometry, considering the projectile's velocity and the launch angle. The formula for the final speed of the projectile when it is about to strike the ground is given by the Pythagorean theorem (horizontal_vel^2 + vertical_vel^2 = final_vel^2), where horizontal_vel is the horizontal_velocity * time and vertical_vel is the square root of 2*g*h. Substituting the calculated values of horizontal and vertical velocities into this formula will give the final velocity.

Evaluate which answers depend on the mass of the projectile

In this problem, both the kinetic and potential energies are directly proportional to the mass of the projectile. Hence, both of these values will change with a change in mass. However, the final velocity of the projectile does not depend on its mass.

Key concepts

Kinetic Energy

Kinetic energy is the energy that a body possesses due to its motion. When a projectile is fired, its kinetic energy can be calculated using the formula: \( KE = \frac{1}{2} \times \text{mass} \times \text{velocity}^2 \). This equation shows that kinetic energy depends not only on the mass of the object but also greatly on the square of its velocity. For the projectile in the exercise, with a mass of 2.4 kg and an initial velocity of 150 m/s, the kinetic energy is significant.

• The greater the velocity, the higher the kinetic energy, since velocity is squared in the equation.
• Mass plays a direct role; doubling the mass will double the kinetic energy.

Understanding kinetic energy helps us predict how much energy is involved when the projectile is set into motion.

Potential Energy

Potential energy is stored energy that depends upon the position of an object relative to a force. When dealing with a projectile fired from a height, gravitational potential energy is considered. The formula used is \( PE = \text{mass} \times g \times \text{height} \), where \(g\) is the acceleration due to gravity and is roughly 9.81 m/s² on Earth.

• Height determines potential energy: higher heights increase the potential energy.
• Like kinetic energy, potential energy is directly proportional to mass; doubling the mass will double the potential energy.

In the provided exercise, the potential energy of the projectile depends on its mass and the height of the cliff (125 m). Calculating potential energy helps us understand the energy available due to the object's elevation above the ground.

Velocity Calculation

Velocity calculation involves determining the speed and direction of an object. The exercise requires finding the speed just before the projectile hits the ground after being fired. This is affected by its initial velocity and the height from which it was launched. The initial velocity can be broken down into horizontal and vertical components using trigonometry:

• Horizontal Velocity: \( v_{x} = v \times \cos(\theta) \)
• Vertical Velocity: \( v_{y} = v \times \sin(\theta) \)

Here, \(v\) is the initial velocity (150 m/s) and \(\theta\) the launch angle (41°). Calculating final velocity involves considering the effects of gravity on the vertical component, using the Pythagorean theorem to find the combined speed just before impact. This shows how velocity changes during flight and is crucial for determining the projectile's behavior.

Trigonometry in Physics

Trigonometry plays a vital role in understanding projectile motion. It helps break down initial velocity into horizontal and vertical components, which are easier to manage mathematically. This is particularly useful in physics problems, like the one in the exercise.Using the trigonometric functions:

• \( \cos \theta\) to find horizontal components.
• \( \sin \theta\) to calculate vertical components.

For the projectile, the angle of launch affects how these components are calculated. Accurately using trigonometry ensures correct application of other physics principles, such as conservation of energy and motion equations. It simplifies complex problems by converting them into approachable calculations.