A student throws a ball vertically upward such that it travels 7.5 to its maximum height.If the ball is caught at the initial height 2.3 after being thrown, what is the ball’s average speed?
What is the ball’s average velocity?
The average velocity is zero, since it returns to the place from which it was thrown.
It travels H = 5.2 m up and 5.2 m down.
It must have been thrown at initial speed
Vo = sqrt(2gH) = 10.1 m/s
The average speed is half that.
To find the ball's average speed and velocity, we need to analyze its motion.
First, let's calculate the time it takes for the ball to reach its maximum height. The vertical motion of the ball can be described using the equation:
y = u*t + (1/2)*a*t^2
Where:
y = vertical displacement (change in height)
u = initial velocity (when the ball is thrown)
t = time taken
a = acceleration due to gravity (-9.8 m/s^2)
Given:
u = 7.5 m/s (upward)
y = -5.2 m (the change in height from 7.5m to 2.3m)
Using the equation, we can rearrange it to solve for time:
y = u*t + (1/2)*a*t^2
-5.2m = 7.5m/s * t + (1/2) * -9.8 m/s^2 * t^2
Simplifying the equation leads to a quadratic equation:
-4.9t^2 + 7.5t - 5.2 = 0
We can solve this equation using the quadratic formula or a numerical method to find the time t. Considering that this equation has two real roots, we select the positive root representing the time it takes to reach the maximum height.
After finding t, we can calculate the time it takes for the ball to reach the height of 2.3m. Add this time to the time required to reach the maximum height to get the total time of flight.
Once we have the total time of flight, we can calculate the average speed and average velocity.
Average Speed:
Average speed is the total distance traveled divided by the total time taken.
To find the total distance traveled, we calculate the distance from the initial height to the maximum height (7.5m) and the distance from the maximum height to the catch height (2.3m). Add these distances together.
Average speed = Total distance traveled / Total time taken
Average Velocity:
Velocity is a vector quantity that considers both magnitude and direction. The displacement from the initial height to the final height (in this case, from 7.5m to 2.3m) is taken into account.
Average velocity = Displacement / Total time taken
To summarize:
1. Calculate the time it takes for the ball to reach its maximum height using the given information and the vertical motion equation.
2. Calculate the total time of flight by adding the time required to reach the height of 2.3m to the time taken to reach the maximum height.
3. Calculate the total distance traveled by adding the distance from the initial height to the maximum height with the distance from the maximum height to the catch height.
4. Calculate the average speed by dividing the total distance traveled by the total time of flight.
5. Calculate the average velocity by dividing the displacement (change in height) by the total time of flight.