At the 2004 Olympic Games in Athens, Dwight
Phillips won the gold medal in men’s long
jump with a jump of 8.59 m. If the angle of
his jump was 23°, what was his takeoff speed?
(Treat the jumper as an object; ignore that his
legs are in front of him when he lands.)
Elena that equation you wrote makes no sense at all......
To find the takeoff speed of Dwight Phillips, we can use the principles of projectile motion. The horizontal distance traveled by a projectile can be calculated using the formula:
Range = (v^2 * sin(2θ)) / g
Where:
- v is the initial velocity or takeoff speed
- θ is the angle of the jump
- g is the acceleration due to gravity
In this case, we know the range, angle, and acceleration due to gravity. We can rearrange the equation to solve for v:
v = √((Range * g) / sin(2θ))
Plugging in the given values:
Range = 8.59 m
θ = 23°
g = 9.8 m/s^2
We can now calculate the takeoff speed using the formula:
v = √((8.59 * 9.8) / sin(2 * 23°))
To find Dwight Phillips's takeoff speed, we need to use the principles of projectile motion. When a projectile (in this case, Dwight Phillips) is in the air, it follows a parabolic trajectory determined by its initial speed and launch angle. Here's how we can solve this problem step by step:
Step 1: Break down the initial velocity into its horizontal and vertical components.
When jumping, the initial velocity can be broken down into two perpendicular components: horizontal (Vx) and vertical (Vy). The vertical component is the takeoff speed we are trying to find, while the horizontal component remains constant throughout the jump.
Step 2: Convert the takeoff angle from degrees to radians.
Since trigonometric functions work with radians, we need to convert the given angle of 23° to radians. To convert, use the formula: angle in radians = angle in degrees * π/180. So, 23° in radians is (23 * π) / 180.
Step 3: Find the vertical component of the initial velocity (Vy).
To find Vy, we can use the formula: Vy = V * sin(angle), where V is the takeoff speed, and angle is the converted angle in radians.
Step 4: Calculate the takeoff speed (V).
Rearranging the formula from step 3, we can solve for V: V = Vy / sin(angle).
Step 5: Substitute the values and calculate.
Substitute the known values into the formula from step 4 and calculate the takeoff speed.
Let's plug in the values and calculate:
Angle (in radians) = (23 * π) / 180
Vy = V * sin(angle)
V = Vy / sin(angle)
Using a calculator, plug in the known values:
Angle (in radians) ≈ 0.4014 (rounded)
V ≈ (8.59 m) / sin(0.4014)
After calculating the expression, you will find the value of V, which is Dwight Phillips's takeoff speed.
L=vₒ²•sin2α/g,
vₒ =sqrt{Lg/ sin2α}=
=sqrt{8.59•9.8/sin46º}=
=10.8 m/s