You stand at the top of a cliff while your friend stands on the ground below you. You drop a ball from rest and see that it takes 0.90 for the ball to hit the ground below. Your friend then picks up the ball and throws it up to you, such that it just comes to rest in your hand.
What is the speed with which your friend threw the ball?
Learning Goal: To learn to read a graph of position versus time and to calculate average velocity.
In this problem you will determine the average velocity of a moving object from the graph of its position as a function of time . A traveling object might move at different speeds and in different directions during an interval of time, but if we ask at what constant velocity the object would have to travel to achieve the same displacement over the given time interval, that is what we call the object's average velocity. We will use the notation to indicate average velocity over the time interval from to . For instance, is the average velocity over the time interval from to .
Find the average velocity over the time interval from 1 to 3 seconds.
Express your answer in meters per second to the nearest integer.
Well, assuming my friend has the ability to throw a ball straight up and catch it effortlessly, I'd say their throwing speed was "out of this world!" After all, it takes some serious skills to perfectly time their throw so that the ball comes to rest in my hand. I think we've found ourselves a future baseball superstar! But unfortunately, without any more information, it's hard to determine the exact speed.
To find the speed with which your friend threw the ball, we can use the concept of conservation of energy.
When the ball is dropped from rest at the top of the cliff, it will gain speed as it falls due to the force of gravity. At the moment it hits the ground, all of its potential energy (mgh) will be converted into kinetic energy (½mv²), where m is the mass of the ball, g is the acceleration due to gravity, and h is the height of the cliff.
Given that it takes 0.90 seconds for the ball to hit the ground, we can use this information to determine the height of the cliff using the equation for free fall:
h = (1/2)gt^2
where g is again the acceleration due to gravity and t is the time taken for the ball to hit the ground. Plugging in the values, we can calculate the height h.
Once we know the height of the cliff, we can find the initial velocity of the ball thrown by your friend required for it to come to rest in your hand when it reaches the top of the cliff. The ball's final kinetic energy will be zero when it comes to rest, but it will have gained potential energy equal to the potential energy it had at the bottom of the cliff, mgh.
Since kinetic energy is given by ½mv², where m is the mass of the ball and v is its velocity, we can equate mgh to ½mv² to solve for the velocity.
Therefore, to find the speed with which your friend threw the ball, you will need to determine the height of the cliff and use that information to calculate the velocity.
The same speed with which it hit the ground.
v = g t
v = 9.8 m/s^2 (.9 seconds) = 8.8 m/s