A long jumper leaves the ground at an angle
of 20.8
◦
to the horizontal and at a speed of
10.3 m/s.
How far does he jump? The acceleration
due to gravity is 9.8 m/s
2
.
Answer in units of m
and
What maximum height does he reach?
Answer in units of
3.593669995 meters
To solve this problem, we can break it down into two parts: the horizontal and vertical components of the long jumper's motion.
First, let's find the horizontal distance the long jumper travels. We know that the initial speed in the horizontal direction is 10.3 m/s and the angle of launch is 20.8 degrees. Since there is no acceleration in the horizontal direction, the horizontal velocity remains constant throughout the jump.
Using trigonometry, we can find the horizontal component of the long jumper's velocity:
Horizontal velocity = initial velocity * cos(angle)
Horizontal velocity = 10.3 m/s * cos(20.8 degrees)
Next, let's find the vertical distance and maximum height reached. To do this, we need to consider the vertical motion of the long jumper.
Using the equation for vertical motion:
Vertical distance = (initial vertical velocity * time) + (0.5 * acceleration due to gravity * time^2)
Since the long jumper leaves the ground, the initial vertical velocity is 0. We can use the equation above to determine the time it takes for the jumper to reach the maximum height.
At the maximum height, the vertical component of the velocity is 0. Using the equation for vertical motion:
0 = (initial vertical velocity) + (acceleration due to gravity * time)
0 = 0 + (9.8 m/s^2 * time)
By solving this equation for time, we can find the time it takes for the long jumper to reach the maximum height.
Once we have the time it takes to reach the maximum height, we can substitute this value back into the equation for vertical distance to find the maximum height.
Now, let's calculate the answers.
Horizontal velocity = 10.3 m/s * cos(20.8 degrees)
Vertical distance = (0 * time) + (0.5 * 9.8 m/s^2 * time^2)
0 = 0 + (9.8 m/s^2 * time)
Vertical distance at maximum height = (0 * time) + (0.5 * 9.8 m/s^2 * time^2)
The horizontal distance traveled by the long jumper is the distance covered in the horizontal direction, which is simply the horizontal velocity multiplied by the total time of flight.
To find the total time of flight, we can use the equation for vertical motion:
0 = (initial vertical velocity) + (acceleration due to gravity * total time of flight)
Since the initial vertical velocity is 0, we can solve this equation for the total time of flight:
Total time of flight = (2 * initial vertical velocity) / acceleration due to gravity
With the total time of flight, we can calculate the horizontal distance:
Horizontal distance = Horizontal velocity * Total time of flight
To find the maximum height reached, we need to consider the vertical motion. At maximum height, the vertical component of the velocity is 0. Using the equation for vertical motion:
0 = (initial vertical velocity) + (acceleration due to gravity * maximum height time)
Since the initial vertical velocity is 0, we can solve this equation for the maximum height time:
Maximum height time = (-initial vertical velocity) / acceleration due to gravity
With the maximum height time, we can calculate the maximum height reached:
Maximum height = (initial vertical velocity * maximum height time) + (0.5 * acceleration due to gravity * maximum height time^2)
Now let's calculate the answers.
Horizontal velocity = 10.3 m/s * cos(20.8 degrees)
Total time of flight = (2 * initial vertical velocity) / acceleration due to gravity
Horizontal distance = Horizontal velocity * Total time of flight
Maximum height time = (-initial vertical velocity) / acceleration due to gravity
Maximum height = (initial vertical velocity * maximum height time) + (0.5 * acceleration due to gravity * maximum height time^2)
By substituting the appropriate values into these equations, we can calculate the answers to the given question.