the impulse-momentum theorem
This theorem answers a simple question: if I push on something, how much does its motion change? The everyday answer is 'a harder or longer push changes the motion more' — and that is almost exactly what the theorem says, made precise.
It states that the net impulse on an object equals its change in momentum: F_net times delta-t = delta-p = m v_f - m v_i, where F_net is the net (total) force, delta-t is how long it acts, m is the mass, and v_i and v_f are the velocities before and after. It is really just Newton's second law rearranged: F_net = m a = m (delta-v / delta-t), so F_net times delta-t = m times delta-v = delta-p. Written this way it handles forces that act only briefly, like a bat striking a ball, using the average force over the contact time.
The theorem is the reasoning behind every cushion, catch, and follow-through. To change an object's momentum by a fixed amount you can use a big force briefly or a gentle force for longer — the product is what matters. Note it is the NET force that counts; if other forces also act, only their total determines the momentum change.
A tennis racket contacts the ball for about 0.005 s and sends a 0.057 kg ball from rest to 50 m/s. The momentum change is 0.057 times 50 = 2.85 kg m/s, so the average force during contact is F = delta-p / delta-t = 2.85 / 0.005 = about 570 N — roughly the weight of a small motorbike, delivered in one blink.
A brief contact plus a large average force adds up to the impulse that flings the ball away.
The theorem uses the average force over the contact time; the instantaneous force during a real impact spikes far higher and then falls. It also requires the NET force — the sum of all forces — not just the one you happen to notice.