Problems dealing with motion are usually the first that students of physics will encounter. Concepts like time, velocity and acceleration are interrelated by formulas that students can rearrange with the help of algebra to apply to different circumstances.

Students can calculate the height of a jump, for instance, from more than one starting point. The height of the jump can be calculated if the acceleration and either the initial velocity or the total time in the air is known.

Write an expression for time in terms of change in velocity, using the formula

*v*_{f} = -*gt* + *v*_{i}

where *v*_{f} is final velocity, *g* is the acceleration due to gravity, *t* is time, and *v*_{i} is initial velocity.

## Time of Flight

Solve the equation for *t*

*t* = (*v*_{f} − *v*_{i})/-*g*

Therefore, the amount of time is equal to the change in velocity divided by the acceleration due to gravity.

## Calculate Time to Reach Highest Point

Calculate the amount of time to reach the highest point of the jump. At the highest point, velocity (*v*_{f}) is zero, so given the initial velocity, the time is

*t* = (*v*_{f} − *v*_{i})/-*g*

Use 9.8 m/s² for the acceleration due to gravity. For example, if the initial velocity is 1.37 m/s, time to reach maximum height is:

*t* = (0 − 1.37)/( − 9.8) = 0.14 s

## Calculate Initial Velocity from Total Time of Flight

The initial velocity *v*_{i} can be calculated using the time to reach the jump height

*v*_{i} = (*gt)*

For example, if the total time is 0.14 seconds:

*v*_{i} = (9.8 × 0.14) *v*_{i} = 1.37 m/s

## Vertical Jump Physics Equation

Calculate the jump height using the formula

*s*_{f} = *s*_{i} + *v*_{i}t _- ½(g_t²)

where *s*_{f} is the final position and *s*_{i} is the initial position. Since jump height is the difference between the final and initial position

*h* = (*s*_{f} − *s*_{i})

simplify the formula to

*h* = *v*_{i}_t - ½(g_t²)

and calculate:

*h* = (1.37 × 0.14) - ½(9.8 × 0.14²) = 0.19 − 0.10 = 0.09 meters

#### Tips

Create your own jump height calculator by programming the jump height formula into your graphing calculator!

#### References

#### Photo Credits

- Ezra Shaw/Digital Vision/Getty Images