Maximal vertical jump height is a common metric used to evaluate the neuromuscular capacity of the lower extremities, and the vertical jump test is the generic test most frequently used for this purpose [16, 22, 39, 44, 52, 55, 71, 72, 87, 102, 104]. The height we can jump from the ground indicates our ability to oppose gravity [11, 89], which explains why the maximal vertical jump is one of the most popular tests for assessing general motor ability and movement performance [1, 5, 12, 15, 19, 21, 44, 45, 71, 76, 87, 89, 93]. However, to exemplify, in the context of soccer, mean jump heights between similar populations have been reported to vary by \(\sim\) 17 cm [40, 101], which far exceeds what could be attributed to variances in movement performance within the same sport. It is important to note that these discrepancies may arise from methodological differences in the studies conducted. Indeed, obtaining a correct measure of jump height is not a straightforward task [102].

According to the laws of Newtonian physics, vertical jump height is determined by the elevation of the center of mass (CoM). Jump height can, therefore, be estimated by tracking the displacement of the CoM throughout a jump, which is exemplified as a countermovement jump (CMJ) in Fig. 1 [24, 61, 71, 87].

A sketch of a countermovement jump (CMJ). A Initial standing still prior to the CMJ. B Countermovement (unloading/braking). C Take-off instant. D Apex of the flight phase. E Landing instant. F Landed after the CMJ. See phase definitions in Refs. [36, 63]. Gray-filled circles represent a graphical estimation of the center of mass

To date, three-dimensional (3D) motion capture systems provide the most direct estimates of CoM kinematics [24, 71, 87]. However, the equipment needed for these measures poses significant environmental constraints, especially for in-field testing of athletes—because they are time-consuming and require expensive equipment and analytical expertise [73, 92, 104]. For these reasons, other, simpler methods to calculate jump height have been developed, such as force platforms [60, 63], contact mats [14], photoelectric cells [32], linear position transducers [21], and smartphone (video/accelerometer) applications [5, 28]. These methods have been developed in addition to even simpler measurements, such as the Sargent test [55, 74, 89]. In their systematic review, Xu et al. [102] concluded that, of all the equipment available for jump height calculations, the force platform is the most appropriate for estimating jump height, as it eliminates some of the errors associated with both the direct and indirect methods listed above. Indeed, the force platform is increasingly popular for jump assessments as it is the only technology that allows direct kinetic analysis of a jump [8, 56, 60, 61, 63]. Moreover, force platform systems are steadily becoming more accessible for testing athletes. They are no longer restricted to the laboratory but exist in portable and more affordable versions, including software that provides instant results. In fact, these trends are mirrored in the literature by a growing number of publications related to the implementation of the force platform for jump assessments in field settings [6, 9, 17, 29, 30, 34, 35, 48, 53, 63,64,65,66, 82, 85, 100, 102].

A force plate is a device that typically measures the 3D ground reaction forces (GRFs), with the vertical component being the most commonly applied in jump-related research (Fig. 2) [60]. The laws of mechanics allow the vertical GRFs to be used to calculate jump height, but different mathematical approaches can be applied [56, 63, 102, 104].

The vertical ground reaction forces (N) measured by a force platform (not depicted) are plotted as the blue solid line against time (s) during a countermovement jump (CMJ). The stick figures represent the time-synchronized movement of a person during the CMJ on the ground (depicted as the dashed gray line)

The fundamental understanding of a jump lies in Newton’s second law of motion,

$$\begin \Sigma } = m \varvec, \end$$

(1)

which states that the sum of forces acting on an object’s CoM, \(\Sigma \varvec\), is equal to its mass, m, multiplied by its acceleration, \(\varvec\). Bold font is here used to indicate a vector quantity. The net acceleration of the CoM from Eq. (1) is

$$\begin \varvec = \frac}, \end$$

(2)

with \(\varvec\) from Eq. (2) being the velocity of the CoM,

$$\begin \varvec = \frac}, \end$$

(3)

where t is time, and \(\varvec\) is the displacement of the CoM. Combining Eqs. (1) and (2), and integrating from the initial, i, to the final, f, state with respect to the movement of interest, yields the impulse–momentum theorem,

$$\begin \int _^ \Sigma \varvec dt = m \varvec_ - m \varvec_. \end$$

(4)

The impulse–momentum theorem describes how the accumulation of force over time (impulse) creates changes in momentum (mass multiplied by velocity), which is fundamental for athletic movements such as vertical jumping. The following sections explain the rationale for applying the impulse–momentum theorem for jump height calculations.

The mechanical energy during a jump is the combination of potential and kinetic energy. Lifting the CoM a vertical distance implies an increase in the potential energy. Potential energy represents the energy due to the position of the CoM in a gravitational field, and the value at the start of the jump is typically chosen as the reference (zero) point for potential energy. Note that the reference point for potential energy can be defined arbitrarily and may vary depending on the context and analyses being performed. In jumping, the reference point for potential energy is often either the participant’s CoM position while standing still or at the take-off position, i.e., the heel-rise makes the difference (Fig. 1). Jump height can thus be defined relative to standing (\(\hbox _}\)) or relative to take-off (\(\hbox _}\)), depending on the reference point applied.

The velocity of the CoM determines the kinetic energy. Since mechanical energy is assumed to be conserved throughout the flight phase, the sum of the potential and kinetic energy is constant,

$$\begin mgz + \frac m w^2 = \text , \end$$

(5)

where z is the vertical coordinate of the CoM, w is the vertical velocity component of the CoM, and g is the gravitational acceleration.

\(\hbox _}\) can be estimated from the take-off velocity by assessment of Eq. (5). During a jump, the CoM moves upward against gravity. After take-off, gravity gradually slows the upward movement until the CoM reaches its highest point of the jump (the apex). At the apex, the vertical kinetic energy of the CoM is zero. This is because the CoM has momentarily stopped moving upwards, and the gravitational potential energy is at its maximum since the CoM has reached its highest position with respect to the reference point. As the CoM descends, potential energy is converted back to kinetic energy. In other words, the mechanical energy remains constant throughout the jump (Eq. (5)). So, to jump as high as possible, the aim will be to increase the velocity of the CoM at take-off. Defining the take-off as point C, the apex as point D (see definitions in Fig. 1), and the difference in the vertical displacement of the CoM between points C and D as h, Eq. (5) simplifies to

$$\begin w_C^2 = w_D^2 + 2 g(z_D-z_C). \end$$

(6)

Defining the jump height as \(h = z_D-z_C\), and solving for \(\hbox _}\), yields

For the purpose of this review, Eq. (7) will be referred to as the take-off velocity method (ToV). To estimate \(\hbox _}\) from Eq. (7), \(w_C\) must be determined from Eq. (4). For convenience, the force is integrated from a state where velocity is zero (e.g., during quiet standing), thus eliminating the last term of Eq. (4). The assumption of zero initial velocity is presumably satisfied when performing the CMJ or the squat jump (SJ) because the participant starts from a quiet standing (CMJ) or squat (SJ) position. When performing a drop jump (DJ), the initial velocity is not zero, as the person drops off a box prior to contact with the force plate. If the drop height is known, the initial velocity can be estimated using Eq. (5). One way of estimating drop height is to simply use box height as a measure of drop height (dashed blue line in Fig. 3) [4, 7, 9, 47, 50, 57, 65, 77]. However, owing to changes in posture (and CoM) during descent, the drop height is not easily approximated by the height of the box [12, 50, 65]. If the box from which the person drops is located on a second force platform (force platform 1 in Fig. 3), the touchdown velocity can be determined analogously to the take-off velocity by using the impulse–momentum theorem (Eq. (5)) [4, 46, 65].

The double force platform method is usually set as a criterion for DJ analyses because obtaining GRF–time data of the jumper prior to the drop (yellow line in Fig. 3) allows for a more accurate estimate of touchdown velocity (Eq. (4)), compared with estimating touchdown velocity on the basis of the box height. With the double force platform procedure, the actual displacement of the CoM as the jumper steps off the box is obtained by double integrating the force data from the force platform from where the person is standing prior to the jump (where the box is located). The displacement of the CoM as the jumper steps off the box is then used to obtain the initial velocity condition through Eq. (5) [4, 12, 20, 23, 65]. Such an approach is, however, practically less feasible as it requires practitioners (and researchers) to possess two force platforms [65]. To circumvent this issue, some researchers have reported DJ heights to be analyzed with the initial velocity of zero taken from the end of the jump (at a period after the landing), and from this, simply applying Eq. (7), integrating backward to take-off [4, 46, 65, 99]. Obtaining a velocity of zero at the end of the jump requires the participants to stand still after they have landed. Alternatively, the flight-time approach can be used (see Sect. 1.4) [4].

Ground reaction forces (GRFs, solid black line) and vertical center of mass (CoM) trajectory (shaded gray area, displacement from standing still on the ground) plotted against time during a drop jump. Force platform 1 measures GRFs through the box, while force platform 2 measures GRFs during the jump and landing. The dashed black line indicates body weight. The dashed blue line illustrates the height of the box from which the person drops (i.e., the “box height”). The distance between the dashed blue and yellow lines indicates the drop in CoM height before contacting force platform 2 (i.e., the “drop height”). The yellow line segment below the axis represents GRFs from standing still to take-off, used with the “ToV: drop height” method. The blue line segment represents GRFs b