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Physics: Density, Vectors, Speed, Acceleration, Velocity

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1.7. Which of the following equations are dimensionally correct? (a) vf = vi + ax
(b) y = (2 m)cos(kx), where k = 2 m-1.

1.26. The radius of a solid sphere is measured to be (6.50 ± 0.20) cm, and its mass is measured to be (1.85 ± 0.02) kg. Determine the density of the sphere in kilograms per cubic meter and the uncertainty in the density.

1. 44. Vector has x and y components of -8.70 cm and 15.0 cm, respectively; vector has x and y components of 13.2 cm and -6.60 cm, respectively. If, what are the components of ?

2. 12. An object moves along the x axis according to the equation x(t) = (3.00t2 - 2.00t + 3.00) m, where t is in seconds. Determine (a) the average speed between t = 2.00 s and t = 3.00 s, (b) the instantaneous speed at t = 2.00 s and at t = 3.00 s, (c) the average acceleration between t = 2.00 s and t = 3.00 s, and (d) the instantaneous acceleration at t = 2.00 s and t = 3.00 s.

2. 25. A truck on a straight road starts from rest, accelerating at 2.00 m/s2 until it reaches a speed of 20.0 m/s. Then the truck travels for 20.0 s at constant speed until the brakes are applied, stopping the truck in a uniform manner in an additional 5.00 s. (a) How long is the truck in motion? (b) What is the average velocity of the truck for the motion described?

2. 55. A rock is dropped from rest into a well. The sound of the splash is heard 2.40 s after the rock is released from rest. How far below the top of the well is the surface of the water? The speed of sound in air (at the ambient temperature) is 336 m/s.

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1.7. Which of the following equations are dimensionally correct? (a) vf = vi + ax
(b) y = (2 m)cos(kx), where k = 2 m-1.

Solution: a) vf = vi + ax

Dimensions of the left hand side: vf (velocity) - L1T-1

Dimensions of the right hand side: vi (velocity) - L1T-1
a (acceleration) - L1T-2
x (displacement) - L1T0
ax - L2T-2
As the dimensions of the two terms on the right hand side are different, the equation is dimensionally incorrect.

(b) y = (2 m)cos(kx), where k = 2 m-1

Dimensions of the left hand side: y (displacement) - L1T0

Dimensions of the right hand side: kx - [L-1T0][ L1T0] = L0T0 (kx is dimensionless)

2m - L1T0
Dimensions of both the sides being same, the equation is correct.

1.26. The radius of a solid sphere is measured to be (6.50 ± 0.20) cm, and its mass is measured to be (1.85 ± 0.02) kg. Determine the density of the sphere in kilograms per cubic meter and the uncertainty in the density.

Solution: In case of multiplication or division of quantities with errors, the resultant quantity has a relative (or percentage) error which is the sum of the relative errors in the quantities being multiplied or divided.

Radius of the solid sphere = R = (6.50 ± 0.20) cm

% error in the radius value = (0.20/6.50) x 100 = 3.07%

Volume of the sphere = V = (4/3) ПR3

As the calculation of volume involves multiplication of the radius R thrice, the error in the volume is 3 x 3.07 = 9.21%

Hence, volume of the sphere V = ...

Solution Summary

The solution determines the density, vectors, speed, acceleration and velocity.

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