State what is meant by a vector quantity.
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a quantity that has both magnitude and direction
Scalars And Vectors
Identify which of the following are vectors: mass, weight, energy, momentum, temperature, acceleration.
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weight; momentum; acceleration (any of the three for full marks); mass / energy / temperature are scalars
Explain the difference between distance and displacement, using a runner who completes one full lap of a 400 m circular track as an example.
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distance is a scalar / has only magnitude; displacement is a vector / has magnitude and direction; distance travelled = 400 m but displacement = 0 because the runner returns to the starting point
Two forces of 8.0 N and 6.0 N act on a point at right angles to each other. Calculate the magnitude of the resultant force.
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$$R=\sqrt{{8.0}^{2}+{6.0}^{2}}$$ $$R=\sqrt{100}=10\text{ N}$$
State the parallelogram rule for adding two coplanar vectors.
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the two vectors are drawn from a common point as adjacent sides of a parallelogram; the resultant is the diagonal of the parallelogram drawn from the common point
A car travels north at 15 m s⁻¹, then turns and travels east at 20 m s⁻¹. Determine the magnitude and direction of the change in velocity.
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Change in velocity = final − initial. Finding the magnitude $$\Delta v=\sqrt{{20}^{2}+{15}^{2}}$$ $$\Delta v=\sqrt{625}=25{\text{ m s}}^{-1}$$ Finding the direction $$\theta ={tan}^{-1}\left(\frac{15}{20}\right)=37^{\circ}$$ The change in velocity is 25 m s⁻¹ at 37° south of east.
Show that the resultant of two equal vectors of magnitude 5.0 N acting at right angles to each other has magnitude 7.07 N.
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$$R=\sqrt{{5.0}^{2}+{5.0}^{2}}=\sqrt{50}$$ $$R=7.07\text{ N as required}$$
A force of 50 N acts at an angle of 60° above the horizontal. Calculate the horizontal and vertical components of this force.
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$${F}_{x}=50cos60^{\circ}=25\text{ N}$$ $${F}_{y}=50sin60^{\circ}=43\text{ N}$$
State why it is useful to resolve a vector into two perpendicular components.
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the two components act independently of each other / motion along one axis is unaffected by motion along the perpendicular axis; this simplifies analysis by reducing a two-dimensional problem to two one-dimensional problems
A child pulls a sledge across level snow with a rope inclined at 25° above the horizontal. The tension in the rope is 80 N. Calculate the horizontal force pulling the sledge forward and the vertical force lifting the sledge.
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Horizontal component $${F}_{x}=80cos25^{\circ}=73\text{ N forward}$$ Vertical component $${F}_{y}=80sin25^{\circ}=34\text{ N upward}$$
A skier of mass 65 kg stands on a slope inclined at 20° to the horizontal. Determine the component of the skier's weight acting parallel to the slope and explain its physical effect.
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Finding the weight $$W=65\times 9.81=638\text{ N}$$ Finding the component parallel to the slope $${W}_{∥}=Wsin20^{\circ}$$ $${W}_{∥}=638\times sin20^{\circ}$$ $${W}_{∥}=218\text{ N}\approx 220\text{ N down the slope}$$ This component is the resultant force component along the slope / it would accelerate the skier down the slope if friction is absent.
A projectile is launched at 30 m s⁻¹ at an angle of 40° above the horizontal. Determine the initial horizontal and vertical components of the velocity, and explain why these components can be analysed separately during the flight.
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Horizontal component $${v}_{x}=30cos40^{\circ}=23{\text{ m s}}^{-1}$$ Vertical component $${v}_{y}=30sin40^{\circ}=19{\text{ m s}}^{-1}$$ The horizontal and vertical motions are independent because the only force acting (gravity) is vertical; gravity causes acceleration only in the vertical direction; therefore the horizontal velocity remains constant throughout flight (ignoring air resistance); the vertical motion is uniformly decelerated then accelerated by g.