The hydroelastic symmetric response of a floating ice sheet caused by a pressure moving either in the ice lead or on the infinite ice sheet with a crack (a lead of zero width) is investigated. The ice sheet is modeled as a viscoelastic thin plate. The water is of constant depth. The flow under the ice is potential and linear. A boundary integral method (BIM) for the flow under the ice is combined with the finite difference method for the ice plate with free-free edge conditions to solve the coupled problem of linear hydroelasticity. Numerical results for deflections and stress distributions are shown to agree well with the available results by others. The proposed approach can be applied to problems with different edge conditions and different positions of the load with respect to the lead. The ice responses are studied with respect to the speed of the load. The speed can be subcritical, critical, and supercritical with respect to the critical speed for a floating infinite elastic plate. The speeds of the load, which provide maximum deflection, maximum stress, and maximum wave-making resistance, are determined. All these speeds are different and greater than the critical speed for an infinite elastic plate. The effect of the ice thickness, lead width, and load properties on these speeds is discussed.