A track star goes in the long jump at 10 m/s at 37 degrees above the horizontal. What is the magnitude of the horizontal displacement?

6m

To find the magnitude of the horizontal displacement, we need to use the horizontal component of the initial velocity.

Given:
Initial velocity (vi) = 10 m/s
Launch angle (θ) = 37 degrees

To find the horizontal component of the initial velocity (vix), we use the equation:
vix = vi * cos(θ)

Calculating:
vix = 10 m/s * cos(37 degrees)

Using a calculator:
vix ≈ 10 m/s * 0.7986

vix ≈ 7.986 m/s

Therefore, the magnitude of the horizontal displacement is approximately 7.986 m/s.

To find the magnitude of the horizontal displacement, we can use a combination of trigonometry and kinematics.

The horizontal displacement can be calculated using the formula:

Displacement = Initial Velocity * Time + (1/2) * Acceleration * Time^2

However, in this case, we are only interested in the horizontal component of the displacement. So we need to find the time it takes for the track star to complete the jump.

To do this, we can use the vertical component of the initial velocity:

Vertical Velocity = Initial Velocity * sin(angle)

where the angle is given as 37 degrees.

Next, we can use kinematics to find the time it takes for the track star to reach the maximum height. At maximum height, the vertical velocity will become zero.

The equation we can use is:

Final Velocity^2 = Initial Velocity^2 + 2 * Acceleration * Displacement

Since the vertical velocity becomes zero at the maximum height, the final velocity becomes 0 m/s. We can plug in the values and solve for the displacement:

0^2 = (Initial Velocity * sin(angle))^2 + 2 * (-9.8 m/s^2) * Displacement

Simplifying the equation gives us:

0 = (10 m/s * sin(37))^2 + 2 * (-9.8 m/s^2) * Displacement

Now we can solve for the displacement:

Displacement = -(10 m/s * sin(37))^2 / (2 * (-9.8 m/s^2))

Calculating the above expression gives us the vertical displacement. To find the horizontal displacement, we can use the equation:

Horizontal Displacement = Initial Velocity * cos(angle) * Time

Substituting the values, we get:

Horizontal Displacement = 10 m/s * cos(37) * Time

Now that we have the time and the horizontal displacement equation, let's find the time first.

To find the time, we can use the equation:

Final Velocity = Initial Velocity + Acceleration * Time

Since the final velocity in this case is equal to the initial velocity in the horizontal direction (no acceleration in the horizontal direction), we can set the equation as:

10 m/s = 10 m/s * cos(37) * Time

Simplifying the equation gives us:

1 = cos(37) * Time

Now we can solve for Time:

Time = 1 / cos(37)

Plugging this value back into the horizontal displacement equation:

Horizontal Displacement = 10 m/s * cos(37) * (1 / cos(37))

Simplifying the equation gives us:

Horizontal Displacement = 10 m/s

Therefore, the magnitude of the horizontal displacement is 10 m/s.