Physics Problem Solving Height And Elastic Constant Calculation

by esunigon 64 views
Iklan Headers

In this article, we will solve a physics problem involving a child jumping on a trampoline. The problem involves calculating the maximum height the child reaches and the elastic constant of the trampoline. To solve this problem, we will use concepts from classical mechanics, including kinematics, energy conservation, and Hooke's law. We will break down the problem step by step, providing clear explanations and calculations. This comprehensive approach will enhance understanding and provide a valuable resource for students and enthusiasts alike.

A child weighing 30 kg jumps onto a trampoline. The child's initial velocity is 6 m/s, and air resistance is 24 N.

A) Calculate the maximum height the child reaches.

B) If the trampoline stretches 30 cm, what is the elastic constant?

To calculate the maximum height the child reaches, we need to consider the forces acting on the child and apply principles of kinematics and energy conservation. The main forces at play here are gravity and air resistance. Gravity acts downward, while air resistance opposes the motion of the child.

Step 1 Mass Weight

First, let's calculate the weight of the child, which is the force exerted on the child due to gravity. The weight (W) can be calculated using the formula:

W = m * g

Where:

  • m is the mass of the child (30 kg).
  • g is the acceleration due to gravity (approximately 9.8 m/s²).

So,

W = 30 kg * 9.8 m/s² = 294 N

Step 2: Net Force

Next, we need to determine the net force acting on the child as they move upwards. The net force (F_net) is the difference between the weight and the air resistance:

F_net = W + F_air

Where:

  • W is the weight of the child (294 N).
  • F_air is the air resistance (24 N).

Since air resistance acts in the opposite direction to the motion (downward), we add it to the weight to find the total downward force:

F_net = 294 N + 24 N = 318 N

Step 3: Acceleration

Now, we can calculate the acceleration (a) of the child using Newton's second law of motion:

F_net = m * a

Where:

  • F_net is the net force (318 N).
  • m is the mass of the child (30 kg).
  • a is the acceleration.

Rearranging the formula to solve for a:

a = F_net / m

a = 318 N / 30 kg = 10.6 m/s²

Since the net force is acting downward, the acceleration is also downward, which means it is deceleration as the child moves upward.

Step 4: Kinematics

To find the maximum height, we can use the following kinematic equation:

v_f² = v_i² + 2 * a * Δy

Where:

  • v_f is the final velocity (0 m/s at the maximum height).
  • v_i is the initial velocity (6 m/s).
  • a is the acceleration (-10.6 m/s², negative because it's deceleration).
  • Δy is the change in height (the maximum height we want to find).

Plugging in the values:

0² = 6² + 2 * (-10.6) * Δy

0 = 36 - 21.2 * Δy

Now, solve for Δy:

21.2 * Δy = 36

Δy = 36 / 21.2

Δy ≈ 1.698 m

Therefore, the maximum height the child reaches is approximately 1.698 meters.

To calculate the elastic constant of the trampoline, we can use Hooke's Law. Hooke's Law relates the force exerted by a spring (or in this case, the trampoline) to the distance it is stretched or compressed.

Step 1: Hooke's Law

Hooke's Law is given by the formula:

F = k * x

Where:

  • F is the force exerted by the spring (in this case, the force exerted by the trampoline).
  • k is the elastic constant (which we want to find).
  • x is the displacement (the distance the trampoline stretches).

Step 2: Force Exerted by the Trampoline

The force exerted by the trampoline is equal to the weight of the child, which we calculated earlier as 294 N. This is because the trampoline is supporting the child's weight when it is stretched.

F = 294 N

Step 3: Displacement

The displacement (x) is the distance the trampoline stretches, which is given as 30 cm. We need to convert this to meters:

x = 30 cm = 0.30 m

Step 4: Solving for the Elastic Constant

Now we can plug the values into Hooke's Law and solve for k:

294 N = k * 0.30 m

k = 294 N / 0.30 m

k = 980 N/m

Therefore, the elastic constant of the trampoline is 980 N/m. This constant indicates the stiffness of the trampoline; a higher value means the trampoline is stiffer.

In this article, we solved a comprehensive physics problem that involved calculating the maximum height a child reaches when jumping on a trampoline and determining the elastic constant of the trampoline. We applied principles from kinematics, Newton's laws of motion, and Hooke's Law. Our calculations showed that the child reaches a maximum height of approximately 1.698 meters and that the trampoline has an elastic constant of 980 N/m. Understanding these concepts is essential for anyone studying physics or related fields. This step-by-step solution not only provides the answers but also illustrates the process of applying physics principles to real-world scenarios. This article should serve as a valuable resource for students and anyone interested in understanding the mechanics of motion and elasticity.

Physics problem, maximum height, elastic constant, trampoline, kinematics, Hooke's Law, Newton's laws of motion, air resistance, initial velocity, acceleration, displacement, force, weight, gravity