The tiny house at Sderot was all but destroyed by the rocket which struck at 8.06am, punching a gaping hole through the roof. Diane Mosadi, a widow of 74 living alone, escaped with only minor injuries. The other homes packed around a narrow alleyway had windows shattered and walls scorched.

The Qassam missile landing on Mrs Mosadi's home was one of 40 fired from Gaza after Israeli armour pushed into the Palestinian territory, starting the ground offensive. Other rockets, including longer-range Grads, hit Netivot, Eshkol, Ashdod and Ashkelon with one salvo hitting Kiryat Malachi, in northern Negev, for the first time.

Just outside Sderot, mortar rounds smashed to the ground about 50 yards from journalists perched on a vantage point at Nir Am. But the real destruction was taking place on the horizon just over the border as Israeli troops and armour carried out a pincer movement either side of Gaza City.

Plumes of smoke rose from Gaza City as it was pounded by artillery and attacked from the air. With machine gun and mortar fire echoing around the hills, helicopter-gunships were scouting the fields.

Around 150 tanks and thousands of troops have fanned out into the Gaza Strip and were engaged in action around the northern town of Beit Lahiya and the Jabaliya refugee camp after securing the former Jewish settlement of Netzarim.

Israeli soldiers moving through the fields and orchards, with their face blackened by camouflage as the offensive began on Saturday evening, had used sniffer dogs in an attempt to detect booby traps. The first few injuries came from mines, but then they reported coming under mortar fire with one of the rounds killing a soldier.

Soon the soldiers were meeting more sustained resistance with sounds of prolonged firefights echoing across the dark fields. Lieutenant-General Gabi Ashkenazi, the Israeli chief of staff, told a cabinet meeting that most of the combat was at close range with the Hamas fighters trying to draw the troops into urban areas.

On the way to Ashdod a cluster of light tanks and armoured personnel carriers had been stationed on the edge of a field. The soldiers said they were on their way to the Gaza Strip. One of them, Sergeant Daniel Galen, said: "We are soldiers and we are following orders ... all I will say is that Hamas has been firing rockets into Israel for eight years and something had to be done to stop it."

In Sderot, Privates Rimat Avigor and Ruth Paz were waiting to see whether they would be allowed to serve in the Gaza Strip.

Nineteen-year-old Private Avigor said: "I come from the north and we had a lot of missile attacks during the Lebanese war. So being here now is not that different."

Yohan Abergil, a 25-year-old student, who lives with his parents in the house opposite the one hit yesterday, pointed to his front door six yards away. "It is a miracle that the old lady who lives here has not been badly hurt. I am very glad we are doing something about Gaza at last, this is a war we must fight."