A playground has its highest point 2.2m above the lowest point, and the slide is 4.9m long. If the average resistance to motion is one-fifth of the weight of the child(mass 40kg), find the speed at which the child reaches the bottom show working?
To find the speed at which the child reaches the bottom of the slide, we can use the principle of conservation of energy. The potential energy at the highest point of the slide is converted into kinetic energy at the bottom.
Step 1: Calculate the potential energy at the highest point.
Potential energy (PE) is given by the formula PE = mgh, where m is the mass of the child (40 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height difference between the highest and lowest points of the playground (2.2 m).
PE = 40 kg * 9.8 m/s² * 2.2 m
PE = 862.4 Joules
Step 2: Calculate the kinetic energy at the bottom.
Kinetic energy (KE) is given by the formula KE = 0.5mv², where m is the mass of the child (40 kg) and v is the velocity at the bottom of the slide.
KE = 0.5 * 40 kg * v²
Step 3: Equate potential energy to kinetic energy.
Since energy is conserved, we can equate the potential energy to the kinetic energy.
PE = KE
862.4 Joules = 0.5 * 40 kg * v²
Step 4: Solve for the velocity.
Rearrange the equation to solve for v.
v² = 2 * 862.4 Joules / 40 kg
v² = 43.12 m²/s²
v = √(43.12 m²/s²)
v ≈ 6.57 m/s
Therefore, the child reaches the bottom of the slide at a speed of approximately 6.57 m/s.