Remember that the velocity in the Y
(vertical) direction (Vy) is equal to the
V*sin(angle). Remember also that HALF
WAY the vertical velocity is 0.
A long jumper leaves the ground at an angle
of 23.1 to the horizontal and at a speed of
9.71 m/s.
How far does he jump? The acceleration
due to gravity is 9.8 m/s
Answer in units of m.
What maximum height does he reach?
Answer in units of m.
u = 9.71 cos 23.1 = 8.93
Vi = 9.71 sin 23.1 = 3.81
Vy = Vi - g t = 3.81 - 9.8 t
at top Vy = 0
then at top 9.8 t = 3.81
t = .389 second at top (vertex of parabolic path)
falls for the same amount of time
so .389 * 2 = .777 second in air
distance = u t = 8.93 * .777 = 6.94 meters long jump
y = Yi + Vi t - 4.9 t^2 where t is time up = .389 s
y = 0 + 3.81 (.389) - 4.9(.389)^2
= 1.48 - .741 = .739 meter
To find the distance the long jumper jumps, we need to calculate the horizontal range. We can use the formula for horizontal range:
Range = (initial velocity * time of flight) * cos(angle)
Where:
- initial velocity is the magnitude of the initial velocity of the jumper, which is given as 9.71 m/s.
- time of flight is the time the jumper spends in the air, which can be calculated using the formula:
time of flight = (2 * initial vertical velocity) / acceleration due to gravity
- acceleration due to gravity is given as 9.8 m/s^2.
- angle is the angle of the jump, given as 23.1 degrees.
To find the maximum height, we need to calculate the vertical displacement. We can use the formula for vertical displacement:
Vertical displacement = (initial vertical velocity^2) / (2 * acceleration due to gravity)
Now let's calculate the values step by step.
Step 1: Calculate the time of flight
time of flight = (2 * initial vertical velocity) / acceleration due to gravity
Given that halfway the vertical velocity is 0, we can use the formula for vertical velocity:
Vy = V * sin(angle)
Since halfway the vertical velocity is 0:
0 = V * sin(angle)
sin(angle) = 0
angle = 0 degrees
Therefore, the vertical velocity halfway is 0, and the time of flight is:
time of flight = (2 * initial vertical velocity) / acceleration due to gravity
time of flight = (2 * V * sin(angle)) / acceleration due to gravity
time of flight = (2 * 9.71 * sin(23.1)) / 9.8
time of flight ≈ 1.46 seconds
Step 2: Calculate the horizontal range
Range = (initial velocity * time of flight) * cos(angle)
Range = (9.71 * 1.46) * cos(23.1)
Range ≈ 11.098 meters
Therefore, the long jumper jumps approximately 11.098 meters.
Step 3: Calculate the maximum height
Vertical displacement = (initial vertical velocity^2) / (2 * acceleration due to gravity)
Vertical displacement = (initial velocity * sin(angle))^2 / (2 * acceleration due to gravity)
Vertical displacement = (9.71 * sin(23.1))^2 / (2 * 9.8)
Vertical displacement ≈ 1.03 meters
Therefore, the long jumper reaches a maximum height of approximately 1.03 meters.