The mass of the helicopter is $9300 \mathrm{~kg}$. It takes off vertically at time $t=0$. The pilot advances the throttle so that the upward thrust of its engine (in $\mathrm{kN}$ ) is given as a function of time in seconds by $T=100+2 t^2$. (a) Determine the magnitude of the linear impulse due to the forces acting on the helicopter from $t=0$ to $t=3 \mathrm{~s}$. (b) Use the principle of impulse and momentum to determine how fast the helicopter is moving at $t=3 \mathrm{~s}$.

An object of mass $m=2 \mathrm{~kg}$ slides with constant velocity $v_0=4 \mathrm{~m} / \mathrm{s}$ on a horizontal table (seen from above in the figure). The object is connected by a string of length $L=1 \mathrm{~m}$ to the fixed point $O$ and is in the position shown, with the string parallel to the $x$ axis, at $t=0$. (a) Determine the $x$ and $y$ components of the force exerted on the mass by the string as functions of time. (b) Use your results from part (a) and the principle of impulse and momentum to determine the velocity vector of the mass at $t=1 \mathrm{~s}$. Strategy: To do part (a), write Newton's second law for the mass in terms of polar coordinates.

A soccer player kicks the stationary $0.45-\mathrm{kg}$ ball to a teammate. The ball reaches a maximum height above the ground of $2 \mathrm{~m}$ at a horizontal distance of $5 \mathrm{~m}$ from the point where it was kicked. The duration of the kick was $0.04$ seconds. Neglecting the effect of aerodynamic drag, determine the magnitude of the average force the player exerted on the ball.

At $t=0$, a 20-kg projectile is given an initial velocity $v_0=20 \mathrm{~m} / \mathrm{s}$ at $\theta_0=60^{\circ}$ above the horizontal. (a) By using Newton's second law to determine the acceleration of the projectile, determine its velocity at $t=3 \mathrm{~s}$. (b) What impulse is applied to the projectile by its weight from $t=0$ to $t=3 \mathrm{~s}$ ? (c) Use the principle of impulse and momentum to determine the projectile's velocity at $t=3 \mathrm{~s}$.

The two crates are released from rest. Their masses are $m_A=20 \mathrm{~kg}$ and $m_B=80 \mathrm{~kg}$. The coefficient of kinetic friction between the contacting surfaces is $\mu_{\mathrm{k}}=0.1$. The angle $\theta=20^{\circ}$. What is the magnitude of the velocity of crate $A$ after $1 \mathrm{~s}$ ?

The two crates are released from rest. Their masses are $m_A=20 \mathrm{~kg}$ and $m_B=80 \mathrm{~kg}$, and the surfaces are smooth. The angle $\theta=20^{\circ}$. What is the magnitude of the velocity of crate $A$ after $1 \mathrm{~s}$ ?

Strategy: Apply the principle of impulse and momentum to each crate individually.

The two crates are released from rest. Their masses are $m_A=40 \mathrm{~kg}$ and $m_B=30 \mathrm{~kg}$, and the coefficient of kinetic friction between crate $A$ and the inclined surface is $\mu_{\mathrm{k}}=0.15$. What is the magnitude of the velocity of the crates after $1 \mathrm{~s}$ ?

The two weights are released from rest at time $t=0$. The coefficient of kinetic friction between the horizontal surface and the 5 - $\mathrm{lb}$ weight is $\mu_{\mathrm{k}}=0.4$. Use the principle of impulse and momentum to determine the magnitude of the velocity of the 10-lb weight at $t=1 \mathrm{~s}$.

Strategy: Apply the principle to each weight individually.

The $100-\mathrm{lb}$ crate is released from rest on the inclined surface at time $t=0$. The coefficient of kinetic friction between the crate and the surface is $\mu_{\mathrm{k}}=0.18$. (a) Determine the magnitude of the linear impulse due to the forces acting on the crate from $t=0$ to $t=2 \mathrm{~s}$. (b) Use the principle of impulse and momentum to determine how fast the crate is moving at $t=2 \mathrm{~s}$.

In a cathode-ray tube, an electron ( mass $=9.11 \times 10^{-31} \mathrm{~kg}$ ) is projected at $O$ with velocity $\mathbf{v}=\left(2.2 \times 10^7\right) \mathbf{1}(\mathrm{m} / \mathrm{s})$. While the electron is between the charged plates, the electric field generated by the plates subjects it to a force $\mathbf{F}=-e E$ J. The charge on the electron is $e=1.6 \times 10^{-19} \mathrm{C}$ (coulombs), and the electric field strength is $E=15 \sin (\omega t) \mathrm{kN} / \mathrm{C}$, where the frequency $\omega=2 \times 10^9 \mathrm{~s}^{-1}$. (a) What impulse does the electric field exert on the electron while it is between the plates? (b) What is the velocity of the electron as it leaves the region between the plates?

The 20-kg package $A$ starts from rest and slides down the smooth ramp. If the hydraulic device $B$ exerts a force of magnitude $F=540\left(1+0.4 t^2\right) \mathrm{N}$ on the package, where $t$ is in seconds measured from the time of first contact, what time is required to bring the package to rest?

In an assembly-line process, the 20-kg package $A$ starts from rest and slides down the smooth ramp. Suppose that you want to design the hydraulic device $B$ to exert a constant force of magnitude $F$ on the package and bring it to rest in $0.15 \mathrm{~s}$. What is the required force $F$ ?

if the crate starts from rest at $t=0$ and the winch exerts a tension $T=1220+200 t \mathrm{~N}$.

The crate has a mass of $120 \mathrm{~kg}$, and the coefficients of friction between it and the sloping dock are $\mu_{\mathrm{s}}=0.6$ and $\mu_{\mathrm{k}}=0.5$. The crate starts from rest, and the winch exerts a tension $T=1220 \mathrm{~N}$. (a) What impulse is applied to the crate during the first second of motion? (b) What is the crate's velocity after $1 \mathrm{~s}$ ?