Derivations and Proportional reasoning
How many times faster would you have to jump in order to jump twice as high? Show your work.
Sqrt(2)*V0
You throw a football 10 m/s @ 40 degrees above the horizontal. Calculate the initial horizontal and vertical velocities.
Vx=7.66 m/s
Vy=6.43 m/s
Draw the horizontal and vertical graphs (x vs t, v vs t, a vs t, y vs. t, v vs. t, a vs. t) for a horizontal launch.
Dr. One will check
What is the velocity of a ball at its peak in a projectile?
The velocity at the peak of a projectile is equal to the horizontal component of the initial velocity (not zero!!!)
A person who can jump a height h has a hangtime of t seconds. Determine how high the person would have to jump for the hangtime to be doubled.
4 x the original height
A person jumps straight up with an initial velocity of 2.45 m/s. The person then returns to the ground. Calculate max height, hangtime, and acceleration.
maximum height: 0.31 m
Hangtime = 0.5 seconds
acceleration = -9.8 m/s^2
Draw the horizontal graphs (x vs t, v vs t, a vs t) for an asymmetrical projectile problem where the starting height is LOWER than the ending height.
All constant velocity! No accel.
You throw a ball horizontally the instant that you drop a different ball from the same height. Which ball hits the ground first? Why?
Both balls hit the ground at the same time. Time is impacted by vo, change in y, and ay. The horizontal launch and dropped ball all experience voy = 0, acceleration of -9.8 m/s2, and the same distance. Therefore, they hit at the same time.
Hang Time Lab: In terms of total time (T) and g, derive an equation for the maximum height.
Go half way to get max height (so t/2)! Use 2nd half of motion so vo = 0.
H=1/8gt2
A marble rolls off a 1.2 meter tall table in a horizontal launch going 3.0 m/s. Determine the marble’s velocity (magnitude and direction), as the marble hits the ground.
5.7 m/s at 58.26 degrees below horizontal
Draw the vertical velocity vs. time and horizontal velocity vs. time for a symmetrical launch (launched 30 degrees above horizontal)
vertical velocity vs. time = negative slope (starts above t-axis and crosses t-axis)
horizontal velocity vs. time = constant velocity!
You throw a ball horizontally the instant that you drop a different ball from the same height. Which ball is moving faster when it hits the ground? Why?
Both balls have an initial vertical velocity of 0. The ball thrown horizontally has an initial horizontal velocity. Because both balls experience the same acceleration and fall the same distance, the ball that is thrown horizontally is moving faster due to a final speed that takes into account both x and y direction.
Hang Time Lab: In terms of total time (T) and g, derive an equation for initial throwing speed.
Use change in y = 0, and change y = vot + 1/2at2
V0= -1/2 gt2
A football is kicked from the ground with a speed of 25 m/s at 40° above the horizontal. The crossbar is 5.0 meters above the ground and 30 meters away. Is the field goal good? If yes, by how much does the ball clear the crossbar? If no, how short is the football?
Yes! The ball will clear the crossbar by 8.15 meters
Draw the vertical graphs (y vs t, v vs t, a vs t) for an asymmetrical projectile problem where the starting height is LOWER than the ending height.
Dr. One will check.
A ball is launched upward at an angle from the ground. Describe the balls horizontal and vertical motion on the way up to the peak, at the peak, and on the way back down to the ground.
Horizontal: Motion is constant- moving forward at a constant rate the entire time
Vertical- slowing down moving up, Vy=0 but changing direction (@ peak), speeding up moving downwards
Derive an equation for the horizontal range of a projectile in terms of V0, θ, and g
x=(-2V02cosθsinθ)/g
A person kicks a soccer ball with an initial velocity of 3.14 m/s at an angle of 27 degrees. The goal is to land the soccer ball on top of the table that is 15 meters away. Solve for the height of the table!
y = 133.42 m