1. A ball moving horizontally clears a net 2 meters high. What maximum speed can it have if it is to fall within 5 meters of the net?
2.A pulley system lifts a load of 800 Newtons, 20 centimeters while the rope pulling it moves 100 cm.
a. How much force must be applied to lift the load?
b. What is the mechanical efficiency of this system if the actual applied force is 200 N?
3. The braking distance for a car traveling 20 mph is 20 m (slippery). What is the braking distance if you are traveling 60 mph?
4. A 1 kg cart starting from rest slides down an incline starting from 1.5 meters high. (The incline is 2.5 m long.) Assuming a frictionless incline,
a. What is the speed (value of the velocity) at that point?
b. what is the kinetic energy of the cart when it has slide down to a point 1 meter high?
1. The maximum speed the ball can have is...wait for it...not too fast! We don't want it going over the net and into outer space. Let's keep it within a reasonable range, shall we?
2. a. To lift the load, you'll need the force of a thousand suns...or maybe just 800 Newtons. Just grab hold of that rope and give it a good yank!
b. Ah, mechanical efficiency, the elusive beast. With an applied force of 200 N, this system has an efficiency of... let me carry the one... 25%. So, it's like giving an Olympic athlete a piggyback ride while they run a marathon.
3. Now, if your car is traveling at 60 mph and you need to brake, give it some space! The braking distance will be, uh, *squints at calculations* 60 meters. So make sure to keep a safe distance, unless you enjoy seeing your bumper become one with the car ahead.
4. a. As the 1 kg cart slides down the incline, it'll pick up speed like a downhill skier on steroids. At that point, its speed (or velocity, if we're being fancy) can be calculated using some magical formulas. Trust me, it's moving pretty briskly!
b. When the cart has slid down to a point 1 meter high, its kinetic energy will be... drumroll, please... 4.9 Joules! That's enough energy to power a miniature disco party on wheels. Dancing cart, anyone?
1. To find the maximum speed of the ball, we can use the principles of projectile motion. The ball will follow a parabolic trajectory. We need to calculate the initial velocity at which the ball needs to be launched in order to reach a height of 2 meters and still fall within 5 meters of the net.
Let's assume the angle of projection is θ. The initial vertical velocity component will be given by V_y = V*sin(θ), where V is the initial velocity of the ball.
To reach a height of 2 meters, the time of flight can be calculated using the equation: time = 2*(V_y) / g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Next, to determine the maximum horizontal distance covered by the ball, we can use the equation: distance = V_x * time, where V_x is the initial horizontal velocity component, which is V*cos(θ).
Given that the maximum distance should be 5 meters, we can substitute V_x = V*cos(θ) and time = 2*(V*sin(θ))/g into the equation: 5 = V_x * (2*(V*sin(θ))/g), and solve for V.
By rearranging the equation, we get V = (5*g) / (2*sin(θ)*cos(θ)), where θ is the angle of projection.
2. a. The force required to lift the load in a pulley system can be calculated using the principle of work. The work done on an object is equal to the force applied multiplied by the distance moved in the direction of the applied force. In this case, the work done on the load is equal to the weight of the load multiplied by the distance it has been lifted.
So, the force required can be calculated using the equation: force = (work done) / (distance).
In this scenario, the work done is equal to the weight of the load multiplied by the vertical distance it has been lifted: work done = weight * vertical distance.
Therefore, force = (weight * vertical distance) / (distance).
Given that the load is 800 Newtons, the vertical distance is 20 centimeters (0.2 meters), and the distance moved by the rope is 100 centimeters (1 meter), we can substitute these values into the equation to find the force required.
b. Mechanical efficiency is the ratio of useful work output to the total work input. It is calculated by dividing the useful work output by the total work input and multiplying by 100% to express it as a percentage.
In this case, the useful work output is the weight lifted multiplied by the vertical distance moved by the load: useful work output = weight * vertical distance.
The total work input is the force applied (actual applied force) multiplied by the distance moved by the rope: total work input = force applied * distance moved.
Therefore, the mechanical efficiency can be calculated using the equation: efficiency = (useful work output / total work input) * 100%.
Given that the actual applied force is 200 N, we can substitute this value into the equation to find the mechanical efficiency.
3. The braking distance of a car depends on various factors, including the speed at which it is traveling. The braking distance is directly proportional to the square of the car's speed.
To calculate the braking distance at 60 mph (miles per hour), we need to consider the relationship between mph and meters per second (m/s). 1 mph is equal to 0.447 meters per second.
Given that the braking distance at 20 mph is 20 meters, we can set up a proportion to find the braking distance at 60 mph:
(20 mph / 60 mph) = (20 m / x), where x represents the unknown braking distance at 60 mph.
By cross-multiplying and solving for x, we can find the braking distance in meters when the car is traveling at 60 mph.
4. a. To determine the speed (velocity) of the cart when it reaches a height of 1.5 meters, we can use the principle of conservation of mechanical energy.
The initial potential energy (mgh) of the cart is converted into its final kinetic energy (0.5mv^2), where m is the mass, g is the acceleration due to gravity, h is the height, and v is the velocity.
Setting the initial potential energy equal to the final kinetic energy, we get:
mgh = 0.5mv^2.
Simplifying the equation by canceling out the mass, we have:
gh = 0.5v^2.
Solving for v, we get:
v = √(2gh).
Given that the height (h) is 1.5 meters and the acceleration due to gravity (g) is approximately 9.8 m/s^2, we can substitute these values into the equation to find the velocity (speed) at that point.
b. The kinetic energy of the cart can be calculated using the equation:
kinetic energy = 0.5mv^2.
Given that the mass (m) is 1 kg and the velocity (v) is the speed found in part a, we can substitute these values into the equation to find the kinetic energy of the cart at a height of 1 meter.