An athlete executing a long jumper leaves the ground at a 27.0 degree angle and travels 7.89 m. What was the takeoff speed? If this speed were increased by just 5.0%, how much longer would the jump be?

Question

An athlete executing a long jumper leaves the ground at a 27.0 degree angle and travels 7.89 m. What was the takeoff speed? If this speed were increased by just 5.0%, how much longer would the jump be?

Answer 1

Let the angle with the horizontal be φ.

and u the take-off speed (m/s)
t = airborne duration (seconds)

The vertical initial velocity, uy
= u sin(φ)
Horizontal velocity, ux
= u cos(φ)

When she lands, vertical distance=0:
0=uy t - (1/2)gt²
t=2(uy)/g
=2u sin(φ)/g

Horizontal distance (with constant velocity ux)
= ux t
= u cos(φ) * 2u sin(φ)/g
= u²sin(2φ)/g

Can you take it from here?

Answer 2

Yes. I can handle it from here. I just needed a place to start. Thanks MathMate!

Answer 3

You're welcome!

Answer 4

To find the takeoff speed of the athlete executing the long jumper, we can start by analyzing the given information. We know the angle at which the athlete leaves the ground (27.0 degrees) and the distance traveled (7.89 m).

First, we can use the given angle and distance to determine the vertical component of the athlete's takeoff velocity. The vertical component can be calculated using the formula:

Vertical component = Distance traveled * sin(angle)

Plugging in the values, we get:

Vertical component = 7.89 m * sin(27.0 degrees)

Next, we can calculate the horizontal component of the takeoff velocity using the formula:

Horizontal component = Distance traveled * cos(angle)

Substituting in the values, we get:

Horizontal component = 7.89 m * cos(27.0 degrees)

Now, we can use the vertical and horizontal components of the takeoff velocity to find the overall magnitude (speed) of the takeoff velocity using the Pythagorean theorem:

Speed = sqrt((Vertical component)^2 + (Horizontal component)^2)

Calculating the speed, we have:

Speed = sqrt((Vertical component)^2 + (Horizontal component)^2)

Once we have the takeoff speed, we can calculate how much longer the jump would be if the speed were increased by 5.0%. To do this, we multiply the original distance traveled by 1.05 (1 + 5.0%) to get the new distance:

New distance = Original distance * (1 + 5.0%)

Finally, we can subtract the original distance from the new distance to find how much longer the jump would be:

Increase in distance = New distance - Original distance

By following these steps, we can determine both the takeoff speed and the increase in distance if the speed were increased by 5.0%.