Long Jump (2d kinematics)

In summary, to find the takeoff speed of a long jumper who leaves the ground at an angle of 23 degrees and travels 8.7m before landing, you can use trigonometric functions to resolve the takeoff velocity into its horizontal and vertical components. From there, you can set up a pair of simultaneous equations and solve for the takeoff speed. Alternatively, you can also use trig functions to find the maximum height the jumper will go and the time it takes to complete the jump, and then use a kinematics equation to solve for the initial velocity. The correct answer for this problem is 11 m/s.
  • #1
Lil'Physicist
2
0

Homework Statement



A long jumper leaves the ground at an angle of 23 degrees and travels through the air for a horizontal distance of 8.7m before landing. What is the takeoff speed of the jumper?

Homework Equations



Trigonometric functions
?

The Attempt at a Solution



Recently started physics and am going over stuff from the last two tests. Got stuck on this one and needed some help.

None of the three standard kinematics equations allows me to solve this (I end up having two unknowns for any equation I choose), and trigging out the velocity in the beginning leaves me with a triangle that only has angle measures. How can I start on this problem?
 
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  • #2
Denoting the takeoff velocity as v, resolve it into the horizontal and vertical components. You will be able to get an expression each for the horizontal and vertical components, which form a pair of simultaneous equations.
 
  • #3
Hi Lil'Physicist. welcome to PF. It is a vey common problem in projectile motion. Open any reference book or visit Hyper Physics site and go through the projectile motion.
Try to find out two expression. (1) time of flight and (2) range of the projectile.
Then try to solve the problem.
 
  • #4
Fightfish said:
Denoting the takeoff velocity as v, resolve it into the horizontal and vertical components. You will be able to get an expression each for the horizontal and vertical components, which form a pair of simultaneous equations.


I ended up with sin 23=v(y-axis)/v for the y-axis velocity and cos 23=v(x-axis)/v for the x-axis velocity. I do not understand which kinematics equation to substitute this into because of the two unknown variables.

If someone could show the first few steps of the problem, I need that to get the ball rolling. The correct answer is 11 m/s... now just to figure out how that is so...
 
Last edited:
  • #5
Lil'Physicist said:
I ended up with sin 23=v(y-axis)/v for the y-axis velocity and cos 23=v(x-axis)/v for the x-axis velocity. I do not understand which kinematics equation to substitute this into because of the two unknown variables.

If someone could show the first few steps of the problem, I need that to get the ball rolling. The correct answer is 11 m/s... now just to figure out how that is so...

Using trig functions, you can find the maximum height the jumper will go...

tan23=(height of jump)/8.7

Now you have height of jump, you can find the time it takes to complete the jump. Once you have time, you have horizontal distance, time and then solve for Vi via equation:
Xf = Vicos(23) * (time) where Xf is the horizontal distance of the jump and Vi is the initial velocity of the jumper
 

What is the definition of long jump?

Long jump is a track and field event in which an athlete runs down a runway and jumps as far as possible over a designated distance. The athlete must take off from a designated board and land on the ground in a sand-filled pit.

What are the key elements of long jump?

The key elements of long jump include speed, technique, and strength. Athletes must have a fast running speed to generate momentum for the jump. Proper technique, including the approach, takeoff, and landing, is crucial for a successful jump. Strength is also important for generating power during the takeoff phase.

What is the role of 2D kinematics in long jump?

2D kinematics is the study of motion in two dimensions, specifically the horizontal and vertical components of an object's motion. In long jump, 2D kinematics is used to analyze the athlete's speed and trajectory during the approach and takeoff phases, as well as the height and distance of the jump.

How does the angle of takeoff affect the distance in long jump?

The angle of takeoff, or the angle at which the athlete leaves the ground, significantly affects the distance in long jump. Too steep of an angle can result in a shorter jump, while too shallow of an angle can lead to a loss of height and distance. The optimal angle of takeoff is typically between 20-30 degrees.

What factors can influence the outcome of a long jump?

Several factors can influence the outcome of a long jump, including wind speed and direction, the condition of the runway and sand pit, and the athlete's physical and mental state. These variables can impact an athlete's speed, technique, and overall performance during the jump.

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